#345561

order	512
nilpotency class	3
isoclinism	isoclinic class of #333271
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

572 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:

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

Polycyclic presentation:

2^9|a^2=efhi,b^2=ehi,c^2=h,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 252757);