#7968619

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

5810 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7875815);