#7779700

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

4633 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

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, 7686896);