#7535762

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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	#44936	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1824 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7442958);