#7045688

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1339 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6952884);