#7739455

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

4974 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)(11,15)(12,13)(14,16)(17,18,23,26)(19,24,20,21)(22,25,30,32)(27,31,28,29);(1,3,5,8)(2,6,4,7)(17,19,23,20)(18,21,26,24)(22,28,30,27)(25,31,32,29);(10,13)(14,15)(19,20)(21,24)(27,28)(29,31);(1,2,5,4)(3,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,20,23,19)(18,24,26,21)(22,28,30,27)(25,31,32,29);(1,2,5,4)(3,6,8,7)(9,11)(10,14)(12,16)(13,15)(17,21,23,24)(18,19,26,20)(22,29,30,31)(25,27,32,28);(1,4,5,2)(3,7,8,6)(17,22,23,30)(18,25,26,32)(19,27,20,28)(21,29,24,31)

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=g,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=dhi,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7646651);