#123429

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₄²
commutator	C₂×C₄	C₄³
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂³	#376
prime radical	C₂⁵	#41
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1790 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)(5,6)(7,8)(9,10)(11,21,24,31)(12,17)(13,18)(14,20)(15,19)(16,23,30,25)(22,35,37,40)(26,32)(27,34)(28,33)(29,36,39,38);(2,4)(9,11)(10,16,19,30)(12,22)(13,23)(14,25)(15,24)(17,29,33,39)(18,21,20,31)(26,36)(27,38)(28,37)(32,35,34,40);(6,8)(9,12,14,27,15,28,13,26)(10,17,20,34,19,33,18,32)(11,22,25,38,24,37,23,36)(16,29,31,40,30,39,21,35)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=i,h^2=i,b^a=bd,c^a=ce,g^a=gi,f^b=fi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30625);