#7620353

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518992
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1397 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)(4,8)(5,6)(10,14)(11,12)(17,18,23,28)(19,24,30,32)(20,25,22,27)(21,26,29,31);(2,6)(7,8)(9,10)(11,16)(12,15)(13,14)(17,19,23,30)(18,24,28,32)(20,29,22,21)(25,31,27,26);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,20,23,22)(18,25,28,27)(19,29,30,21)(24,31,32,26);(1,4,5,3)(2,8,6,7)(9,11,13,12)(10,15,14,16)(17,21,22,19,23,29,20,30)(18,26,27,24,28,31,25,32);(2,6)(7,8)(9,12,13,11)(10,15,14,16)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,d^2=fgh,e^2=g,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

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, 7527549);