#7606650

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1254 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=f,c^2=ghi,d^2=g,e^2=fi,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,e^c=ei,e^d=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, 7513846);