#148899

order	512
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#466
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

494 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

(5,6,7,8)(10,15)(11,20)(12,16)(13,24)(17,32)(18,35)(19,29)(21,23)(22,30)(25,31)(26,33)(27,28)(34,38)(36,37);(1,2)(3,4)(9,10,16,28,13,18,25,29,14,19,31,35,24,27,12,15)(11,17,30,39,22,32,20,33,23,34,37,40,36,38,21,26);(2,4)(9,11,12,21,24,36,31,37,14,23,25,20,13,22,16,30)(10,17,15,26,27,38,35,40,19,34,29,33,18,32,28,39)

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=d,d^2=hi,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,c^b=cf,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56095);