#378029

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1163 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=e,d^2=fh,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 285225);