#6352168

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:

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

Polycyclic presentation:

2^9|c^2=hi,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, #72318860
maximal normal subgroup of	#72232826, #72318860

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259364);