#382605

order	512
nilpotency class	3
isoclinism	isoclinic class of #382477
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2310
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14339
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1121 conjugacy classes of subgroups; only the 312 normal subgroups are shown.

normal subgroups only

This group has 1121 conjugacy classes of subgroups — too many to draw. Only its 312 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,f^c=fi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289801);