#357176

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1144 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=gh,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,d^b=de,e^b=ei,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264372);