#512316

order	512
nilpotency class	4
isoclinism	isoclinic class of #511963
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

544 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 154 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=efhi,d^2=e,e^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gh,h^a=hi,c^b=ci,g^d=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419512);