#123881

order	512
nilpotency class	3
isoclinism	isoclinic class of #8603
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#117
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂³	#380
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1586 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,g^b=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31077);