#316152

order	512
nilpotency class	3
isoclinism	isoclinic class of #316149
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

555 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 555 conjugacy classes of subgroups — too many to draw. Only its 158 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

(2,5)(3,7)(6,8)(9,10,16,20,15,22,14,21)(11,23,27,40,28,47,26,17)(12,29,31,44,32,43,36,18)(13,33,37,46,35,45,34,19)(24,41,51,48,53,38,52,65)(25,54,56,67,57,39,55,49)(30,42,59,58,60,66,62,61)(50,69,70,64,72,68,71,63);(2,6)(5,8)(9,11,14,26,15,28,16,27)(10,17,20,23,22,40,21,47)(12,24,31,51,32,53,36,52)(13,25,34,55,35,57,37,56)(18,41,43,65,44,38,29,48)(19,39,33,49,46,54,45,67)(30,50,59,70,60,72,62,71)(42,64,61,69,66,63,58,68);(1,2,3,5,4,6,7,8)(9,12,15,32)(10,18)(11,24)(13,30,35,60)(14,31,16,36)(17,38,40,41)(19,42)(20,43)(21,29)(22,44)(23,48,47,65)(25,50)(26,51)(27,52)(28,53)(33,61)(34,59,37,62)(39,63,54,64)(45,58)(46,66)(49,68,67,69)(55,70)(56,71)(57,72);(9,13,16,37,15,35,14,34)(10,19,21,45,22,46,20,33)(11,25,27,56,28,57,26,55)(12,30,36,62,32,60,31,59)(17,39,47,67,40,54,23,49)(18,42,29,58,44,66,43,61)(24,50,52,71,53,72,51,70)(38,63,65,69,41,64,48,68)

Polycyclic presentation:

2^9|a^2=egi,b^2=e,c^2=ef,d^2=eh,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,f^b=fh,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 223348);