#6354260

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354237
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1647 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,10)(4,9)(5,13)(6,8)(7,11)(12,16)(14,15)(17,18,22,25)(19,21,24,31)(20,27,30,26)(23,28,32,29);(17,19,21,25,22,24,31,18)(20,26,28,32,30,27,29,23);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,20)(18,23)(19,26)(21,28)(22,30)(24,27)(25,32)(29,31);(2,8)(7,15)(9,13)(10,16)(17,21,22,31)(18,19,25,24)(20,28,30,29)(23,26,32,27);(1,4,6,5)(2,9,8,13)(3,11,12,14)(7,16,15,10)(17,21,22,31)(18,24,25,19)(20,29,30,28)(23,26,32,27)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306700, #72306703, #72306705, …
maximal normal subgroup of	#72306700, #72306703, #72306705, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6261456);