#6816542

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1687 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=gh,d^2=f,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6723738);