#287224

order	512
nilpotency class	3
isoclinism	isoclinic class of #281531
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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

654 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)(17,18,23,28,21,26,22,27)(19,29,31,32,20,24,30,25);(2,5)(6,7)(10,13)(11,15)(14,16)(17,19,22,30,21,20,23,31)(18,24,27,32,26,29,28,25);(9,10,12,14)(11,16,15,13)(17,20,21,19)(18,25,26,32)(22,30,23,31)(24,27,29,28);(1,3,4,8)(2,6,5,7)(18,26)(24,29)(25,32)(27,28)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 194420);