#6366964

order	512
nilpotency class	3
isoclinism	isoclinic class of #6366870
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#3544	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1462 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=hi,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,d^b=df,e^b=ehi,f^b=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306988, #72306999, #72307019
maximal normal subgroup of	#72306988, #72306999, #72307019

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6274160);