#8433758

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

5208 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8340954);