#7648459

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

4784 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)(12,14)(13,16)(17,18)(19,24)(20,26)(21,23)(22,32)(25,31)(27,28)(29,30);(1,3,5,8)(2,6,4,7)(9,11,14,13)(10,15,12,16)(17,19,23,30)(18,24,21,29)(20,28,31,22)(25,32,26,27);(9,12,14,10)(11,15,13,16)(17,20)(18,25)(19,28)(21,26)(22,30)(23,31)(24,32)(27,29);(9,13,14,11)(10,15,12,16)(17,21,23,18)(19,29,30,24)(20,26,31,25)(22,32,28,27);(1,4,5,2)(3,7,8,6)(17,20)(18,26)(19,28)(21,25)(22,30)(23,31)(24,27)(29,32);(17,22,23,28)(18,27,21,32)(19,20,30,31)(24,25,29,26)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=i,d^2=h,e^2=g,f^2=hi,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, 7555655);