#148900

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

Polycyclic presentation:

2^9|a^2=g,b^2=dhi,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, 56096);