#6674539

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 ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1960 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)(7,15)(8,12)(9,14)(10,13)(11,17)(16,20)(18,19)(21,22)(23,28)(24,34)(25,30)(26,35)(27,31)(29,33)(32,36);(1,2)(3,4)(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,15)(14,20)(21,23,27,24)(22,28,31,34)(25,33,26,32)(29,35,36,30);(2,4)(5,8,10,9)(6,12,13,14)(7,16,17,18)(11,19,15,20)(21,24,27,23)(22,29,31,36)(25,33,26,32)(28,35,34,30);(6,13)(11,15)(12,14)(19,20)(21,25,27,26)(22,30,31,35)(23,32,24,33)(28,36,34,29);(2,4)(5,9,10,8)(6,14,13,12)(7,16,17,18)(11,19,15,20)(21,25,27,26)(22,30,31,35)(23,33,24,32)(28,29,34,36)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308006, #72308118, #72308443
maximal normal subgroup of	#72308006, #72308118, #72308443

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6581735);