#129524

order	512
nilpotency class	4
isoclinism	isoclinic class of #12741
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1786
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8310
prime radical	C₂²×#135	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

795 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,d^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dg,g^b=gi,f^c=fi

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, 36720);