#6352188

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

3146 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)(4,16)(6,19)(10,11)(12,27)(13,28)(14,22)(20,23)(21,24)(25,30)(26,31)(29,32);(1,2)(3,10)(4,11)(5,12)(6,22)(7,14)(8,13)(9,19)(15,29)(16,31)(17,26)(18,30)(20,25)(21,28)(23,27)(24,32);(1,3,15,5)(2,10,29,12)(4,6,23,21)(7,17,8,18)(9,25,32,31)(11,22,27,28)(13,30,14,26)(16,19,20,24);(1,4)(2,11)(3,6)(5,21)(7,16)(8,20)(9,26)(10,22)(12,28)(13,25)(14,31)(15,23)(17,19)(18,24)(27,29)(30,32);(1,5,15,3)(2,12,29,10)(4,19,23,24)(6,20,21,16)(7,18,8,17)(9,27,32,11)(13,26,14,30)(22,25,28,31)

Polycyclic presentation:

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259384);