#6675684

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1909 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)(7,14)(8,17)(9,12)(10,13)(11,16)(15,19)(18,20)(21,22)(23,27)(24,29)(25,28)(26,31)(30,33)(32,36)(34,35);(1,2)(3,4)(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,14)(17,20)(21,23,26,32)(22,27,31,36)(24,33,34,25)(28,29,30,35);(2,4)(22,28)(23,32)(25,33)(27,35)(29,36)(30,31);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,24)(22,29)(23,33)(25,32)(26,34)(27,30)(28,36)(31,35);(2,4)(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(23,32)(24,34)(27,36)(29,35)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283319, #72283986, #72284523, …
maximal normal subgroup of	#72283319, #72283986, #72284523, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6582880);