#6345076

order	512
nilpotency class	3
isoclinism	isoclinic class of #61049
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	C₂²×#135	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2920 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1776 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=h,f^2=h,g^2=i,b^a=bf,d^a=di,e^a=eh,f^a=fh,f^b=fh,d^c=di,e^c=eh

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, 6252272);