#7038710

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	#31064	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1549 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=f,d^2=i,e^2=fi,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,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, 6945906);