#357179

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

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=dei,e^b=ei,d^c=di,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, 264375);