#7600392

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1409 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)(4,6)(5,8)(17,18)(19,26)(20,24)(21,27)(22,28)(23,25)(29,31)(30,32);(1,3,4,8)(2,5,6,7)(9,10)(11,13)(12,14)(15,16)(17,19,22,27,23,29,20,30)(18,21,24,26,25,32,28,31);(2,6)(3,8)(9,11,12,15)(10,13,14,16)(17,18)(19,21)(20,28)(22,24)(23,25)(26,27)(29,32)(30,31);(2,6)(5,7)(10,14)(13,16)(17,20,23,22)(18,24,25,28)(19,27,29,30)(21,31,32,26);(2,6)(5,7)(17,21,20,31,23,32,22,26)(18,19,28,30,25,29,24,27)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7507588);