#7968164

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

5347 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gh,d^2=gh,e^2=g,f^2=g,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, 7875360);