#7205036

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200002
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1675 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)(3,5)(4,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,18,22,26)(19,23,21,25)(20,28,30,31)(24,29,32,27);(1,3)(2,5)(4,7)(6,8)(9,11,14,10)(12,15,13,16)(17,19,22,21)(18,23,26,25)(20,29,30,27)(24,28,32,31);(3,7)(5,8)(9,12,14,13)(10,15,11,16)(17,20,22,30)(18,24,26,32)(19,27,21,29)(23,31,25,28);(2,6)(3,7)(9,13,14,12)(10,15,11,16)(17,19,22,21)(18,25,26,23)(20,27,30,29)(24,28,32,31);(3,7)(5,8)(10,11)(15,16)(20,30)(24,32)(27,29)(28,31)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fi,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=ehi,e^c=ei,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, 7112232);