#8081083

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

4847 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)(3,5)(4,6)(7,8)(10,14)(15,16)(17,18,23,27)(19,24,28,32)(20,22,26,31)(21,30,29,25);(9,10)(11,15)(12,16)(13,14)(17,19)(18,24)(20,29)(21,26)(22,25)(23,28)(27,32)(30,31);(1,2,4,6)(3,5,7,8)(9,11,13,12)(10,15,14,16)(19,28)(21,29)(24,32)(25,30);(3,7)(5,8)(9,11,13,12)(10,15,14,16)(17,20)(18,22)(19,21)(23,26)(24,30)(25,32)(27,31)(28,29);(1,3)(2,5)(4,7)(6,8)(9,11,13,12)(10,15,14,16)(17,21)(18,25)(19,26)(20,28)(22,24)(23,29)(27,30)(31,32);(1,2)(3,5)(4,6)(7,8)(9,12,13,11)(10,16,14,15)(17,22)(18,26)(19,25)(20,27)(21,24)(23,31)(28,30)(29,32)

Polycyclic presentation:

2^9|a^2=i,c^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,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, 7988279);