#6365649

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
prime radical	C₂×#127	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1459 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=hi,d^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	#72306801, #72306833
maximal normal subgroup of	#72306801, #72306833

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6272845);