#229183

order	512
nilpotency class	3
isoclinism	isoclinic class of #3290
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2361
prime radical	C₂×#41	C₂²×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

635 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=eh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=de,e^b=eh,g^b=gh,f^c=fh

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, 136379);