#7609575

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	#44915	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1821 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,21)(20,25)(22,24)(23,28)(26,29)(27,30)(31,32);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(18,19)(20,27)(22,29)(23,30)(25,31)(28,32);(10,14)(11,15)(17,19,21,18)(20,23,28,25)(22,26,29,24)(27,31,32,30);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(17,20,22,27,21,28,29,32)(18,23,24,31,19,25,26,30);(2,7)(6,8)(18,19)(23,25)(24,26)(30,31)

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,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, 7516771);