#6690743

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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 ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1960 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257352, #72257411, #72264300, …
maximal normal subgroup of	#72257352, #72257411, #72264300, …
maximal quotient of	#72308534

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6597939);