#7712360

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

5233 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)(3,7)(4,5)(6,8)(9,10,14,12)(11,16,13,15)(17,18,23,20)(19,26,25,22)(21,28,29,24)(27,30,31,32);(9,11,14,13)(10,15,12,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,32)(29,31);(2,5)(6,7)(10,12)(15,16)(18,20)(19,25)(21,29)(30,32);(1,2)(3,6)(4,5)(7,8)(9,12,14,10)(11,16,13,15)(17,20,23,18)(19,26,25,22)(21,28,29,24)(27,32,31,30);(1,3,4,8)(2,6,5,7)(9,13,14,11)(10,16,12,15)(17,21,23,29)(18,24,20,28)(19,27,25,31)(22,30,26,32);(17,22,23,26)(18,25,20,19)(21,30,29,32)(24,31,28,27)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7619556);