#6675403

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1834 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)(7,14)(8,17)(9,12)(10,13)(11,16)(15,19)(18,20)(21,22)(23,29)(24,25)(26,28)(27,32)(30,31)(33,36)(34,35);(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,14)(17,20)(21,23,25,32)(22,27,24,29)(26,31,35,33)(28,36,34,30);(2,4)(21,24,25,22)(23,30,32,36)(26,28,35,34)(27,31,29,33);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(22,24)(27,29)(28,34)(30,36);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(23,31)(26,35)(27,36)(28,34)(29,30)(32,33)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256890, #72256919, #72257050
maximal normal subgroup of	#72256890, #72256919, #72257050
maximal quotient of	#72264138, #72265115

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6582599);