#6675356

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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

1964 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:

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

Polycyclic presentation:

2^9|d^2=f,e^2=f,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,h^c=hi,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308484, #72308851
maximal normal subgroup of	#72308484, #72308851

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6582552);