#6352158

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
prime radical	#35692	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3082 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:

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

Polycyclic presentation:

2^9|b^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, #72234734
maximal normal subgroup of	#72232826, #72234734

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259354);