#7509418

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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	#2537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1327 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)(3,4)(5,7)(6,8)(9,10)(11,14)(12,16)(13,15)(17,18)(19,22)(20,27)(21,25)(23,26)(24,30)(28,29)(31,32);(1,3)(2,6)(4,7)(5,8)(9,11,13,12)(10,14,15,16)(17,19,22,18)(20,28,30,32)(21,25,23,26)(24,29,27,31);(9,12,13,11)(10,14,15,16)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,32)(26,28);(2,7)(3,8)(10,15)(14,16)(17,21,22,23)(18,25,19,26)(20,29,30,31)(24,32,27,28);(1,4,5,6)(2,3,7,8)(17,21,22,23)(18,26,19,25)(20,29,30,31)(24,28,27,32)

Polycyclic presentation:

2^9|b^2=g,c^2=gi,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,h^b=hi,h^c=hi

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, 7416614);