#123697

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	#382
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1474 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,e^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,g^b=gi,g^c=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, 30893);