#7589008

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514000
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 ◁ C₄×#126 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1280 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)(11,16)(12,15)(13,14)(18,23)(24,31)(25,30)(26,32);(1,3,5,4)(2,6,8,7)(9,10)(11,16)(12,15)(13,14)(17,18)(19,26)(20,25)(21,23)(22,30)(24,29)(27,32)(28,31);(1,3,5,4)(2,7,8,6)(10,14)(15,16)(19,27)(24,31)(26,32)(28,29);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,15,14,16)(17,19,21,27)(18,24,23,31)(20,29,22,28)(25,32,30,26);(1,4,5,3)(2,7,8,6)(9,12,13,11)(10,15,14,16)(17,20,21,22)(18,25,23,30)(19,28,27,29)(24,26,31,32)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=fi,d^2=fg,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^d=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7496204);