#7142571

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994107
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#126 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1359 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)(3,7,8,6)(9,10,12,16,13,11,14,15)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32);(1,3,5,8)(2,6,4,7)(10,11)(15,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32);(9,11,13,10)(12,16,14,15)(17,19,21,27)(18,22,24,25)(20,28,29,26)(23,31,32,30);(1,2,5,4)(3,7,8,6)(10,11)(15,16)(17,20,21,29)(18,23,24,32)(19,26,27,28)(22,30,25,31);(1,4,5,2)(3,7,8,6)(9,12,13,14)(10,15,11,16)(18,24)(22,25)(23,32)(30,31)

Polycyclic presentation:

2^9|a^2=fghi,b^2=fi,c^2=h,d^2=fh,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,e^b=ehi,d^c=dh,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7049767);