#7699092

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

4762 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7606288);