#6346505

order	512
nilpotency class	3
isoclinism	isoclinic class of #18192
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#120
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3131 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)(3,11)(4,9)(5,8)(6,10)(7,13)(12,16)(14,15)(22,28)(29,32);(1,3)(2,7)(4,13)(5,12)(6,14)(8,11)(9,15)(10,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,28)(27,30)(31,32);(17,19)(18,22)(20,25)(21,26)(23,29)(24,28)(27,31)(30,32);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,15,16,11);(1,4,6,5)(2,9,10,8)(3,13,14,12)(7,15,16,11)(17,20,21,27)(18,23,24,30)(19,25,26,31)(22,29,28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268392, #72268393, #72268401, …
maximal normal subgroup of	#72268392, #72268393, #72268401, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253701);