#7489971

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309080
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

1705 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

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,d^a=di,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

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7397167);