#7007312

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1502 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6914508);