#7254823

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234453
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×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1688 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7162019);