#7490230

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309060
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	#53098	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1619 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=i,d^2=fhi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7397426);