#6368613

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368558
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	#62486	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1445 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306688, #72306699, #72306705
maximal normal subgroup of	#72306688, #72306699, #72306705

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6275809);