#7742907

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

4603 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:

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

Polycyclic presentation:

2^9|a^2=i,e^2=i,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, 7650103);