#7656054

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

4920 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gh,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

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, 7563250);