#6352165

order	512
nilpotency class	3
isoclinism	isoclinic class of #6352157
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3147 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|c^2=h,d^2=h,e^2=h,f^2=h,b^a=bf,c^a=cg,d^a=dh,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232826, #72234738, #72318685
maximal normal subgroup of	#72232826, #72234738, #72318685

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259361);