#8081065

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

5111 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)(9,10,13,12)(11,15,16,14)(17,18)(19,29)(20,26)(21,32)(22,28)(23,25)(24,30)(27,31);(9,11,13,16)(10,14,12,15)(17,19,23,30)(18,24,25,29)(20,31,28,21)(22,27,26,32);(1,2,4,6)(3,5,7,8)(9,10,13,12)(11,14,16,15)(18,25)(19,30)(21,31)(22,26);(3,7)(5,8)(17,20)(18,26)(19,21)(22,25)(23,28)(24,27)(29,32)(30,31);(1,3)(2,5)(4,7)(6,8)(17,21)(18,27)(19,20)(22,29)(23,31)(24,26)(25,32)(28,30);(1,2)(3,5)(4,6)(7,8)(9,12,13,10)(11,15,16,14)(17,22,23,26)(18,28,25,20)(19,27,30,32)(21,29,31,24)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,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, 7988261);