#7535226

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1422 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=gi,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,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, 7442422);