#287358

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	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

628 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=efi,b^2=fi,c^2=h,d^2=e,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, 194554);