#6352064

order	512
nilpotency class	3
isoclinism	isoclinic class of #6352061
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

3185 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)(4,14)(8,19)(9,18)(10,15)(12,25)(16,27)(20,29)(21,26)(22,28)(23,31)(30,32);(1,2)(3,8)(4,15)(5,10)(6,9)(7,14)(11,20)(12,26)(13,21)(16,28)(17,22)(18,27)(19,25)(23,32)(24,30)(29,31);(4,16)(7,18)(12,23)(14,27)(15,28)(19,29)(25,31)(26,32);(1,3,5,13)(2,8,10,21)(4,12,14,25)(6,11,17,24)(7,19,15,26)(9,20,22,30)(16,23,27,31)(18,29,28,32);(3,11)(8,20)(12,23)(13,24)(19,29)(21,30)(25,31)(26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

80 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, 6259260);