#7611067

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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
prime radical	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1447 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=hi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7518263);