#6816524

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806527
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×#555	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1762 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)(21,22,25,28)(23,30,24,27)(26,29,32,35)(31,36,33,34);(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,14)(17,20)(23,31)(24,33)(26,32)(27,34)(29,35)(30,36);(1,2,3,4)(21,23,26,31,25,24,32,33)(22,27,29,34,28,30,35,36);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(21,24,26,33,25,23,32,31)(22,27,29,34,28,30,35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6723720);