#6675780

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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

2006 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281078, #72284625, #72285224, …
maximal normal subgroup of	#72281078, #72284625, #72285224, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6582976);