#503235

order	512
nilpotency class	4
isoclinism	isoclinic class of #503234
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#25771
commutator, Frattini, cosocle	C₂×#41	C₂⁴
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1130 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=f,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^b=gi,g^c=ghi,h^c=hi,g^f=gi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237521, #72237523, #72237552, …
maximal normal subgroup of	#72237521, #72237523, #72237552, …
maximal quotient of	#72312384, #72312475, #72312534, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410431);