#7507770

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23393	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1887 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=f,c^2=g,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,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, 7414966);