#290224

order	512
nilpotency class	3
isoclinism	isoclinic class of #282043
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

577 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=efh,b^2=ef,c^2=h,d^2=eh,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=deh,c^b=cg,g^b=gi,f^c=fh,g^c=ghi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 197420);