#7590715

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1270 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fg,c^2=g,d^2=i,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,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, 7497911);