#6352329

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×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2885 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=hi,c^2=h,d^2=i,e^2=i,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	#72318860
maximal normal subgroup of	#72318860

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259525);