#8070449

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5096 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=gh,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7977645);