#8080771

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

5072 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,d^2=h,e^2=h,f^2=h,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, 7987967);