#495212

order	512
nilpotency class	4
isoclinism	isoclinic class of #2218
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂³	C₂²×D₈
prime radical	C₂²×#508	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=fh,d^2=i,e^2=hi,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,h^a=hi,c^b=ci

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, 402408);