#6815782

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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	C₂×#3321	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1786 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)(8,12)(9,18)(10,13)(11,17)(15,20)(16,19)(21,22,27,30)(23,31,28,25)(24,26,29,34)(32,33,36,35);(5,7)(6,11)(8,15)(9,16)(10,17)(12,19)(13,14)(18,20)(21,23)(22,25)(24,34)(26,29)(27,28)(30,31)(32,35)(33,36);(1,2,3,4)(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,19,14,20)(21,24,25,35,27,29,31,33)(22,26,23,32,30,34,28,36);(5,9,10,8)(6,12,13,18)(7,16,17,15)(11,19,14,20)(21,25,27,31)(22,28,30,23)(24,35,29,33)(26,36,34,32);(7,17)(11,14)(15,16)(19,20)(21,26,31,36,27,34,25,32)(22,29,28,35,30,24,23,33)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6722978);