#6686029

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1726 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^c=ei,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, 6593225);