#6365692

order	512
nilpotency class	3
isoclinism	isoclinic class of #6365522
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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1465 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306971, #72306977, #72307019
maximal normal subgroup of	#72306971, #72306977, #72307019

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6272888);