#6809729

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1660 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:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=h,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=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, 6716925);