#7489907

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309060
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 ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1660 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)(3,6)(4,5)(7,8)(17,18,24,32,23,27,22,26)(19,25,28,31,21,20,29,30);(2,5)(3,7)(10,14)(11,12)(17,19,23,21)(18,25,27,20)(22,29,24,28)(26,30,32,31);(9,10)(11,16)(12,15)(13,14)(17,20,23,25)(18,19,27,21)(22,30,24,31)(26,28,32,29);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,21,23,19)(18,20,27,25)(22,28,24,29)(26,31,32,30);(2,5)(6,8)(9,12,13,11)(10,15,14,16)(17,22,23,24)(18,26,27,32)(19,28,21,29)(20,30,25,31)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,d^2=fh,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

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, 7397103);