#7083538

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1516 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)(9,10)(11,13)(12,14)(15,16)(17,18,23,27)(19,20,25,30)(21,26,22,24)(28,32,31,29);(9,10)(11,15)(12,14)(13,16)(18,24)(19,25)(20,29)(21,22)(26,27)(30,32);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,19,23,25)(18,20,27,30)(21,31,22,28)(24,32,26,29);(11,16)(13,15)(17,20,23,30)(18,25,27,19)(21,29,22,32)(24,31,26,28);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,23,22)(18,24,27,26)(19,28,25,31)(20,29,30,32)

Polycyclic presentation:

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

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, 6990734);