#7487377

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309000
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	#45482	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1430 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=fh,c^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,g^c=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7394573);