#7644699

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

4978 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)(11,14)(12,16)(13,15)(17,18,21,24)(19,22,27,25)(20,28,29,23)(26,31,32,30);(1,3,5,8)(2,6,4,7)(9,11,13,16)(10,14,15,12)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,29)(31,32);(9,11)(10,12)(13,16)(14,15)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,29)(31,32);(9,12)(10,11)(13,14)(15,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,29)(30,32);(1,2,5,4)(3,6,8,7)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,30,28,31);(1,4,5,2)(3,7,8,6)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7551895);