#8464350

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

4948 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)(3,4)(5,7)(6,8)(10,13)(11,15)(18,24)(19,29)(20,28)(27,31);(1,3,6,7)(2,4,8,5)(9,10,12,13)(11,14,15,16)(17,18,23,24)(19,25,29,32)(20,22,28,26)(21,27,30,31);(1,4,6,5)(2,7,8,3)(9,10)(11,14)(12,13)(15,16)(17,19,23,29)(18,25,24,32)(20,30,28,21)(22,31,26,27);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,20)(18,26)(19,30)(21,29)(22,24)(23,28)(25,27)(31,32);(9,11)(10,14)(12,15)(13,16)(17,21,23,30)(18,27,24,31)(19,20,29,28)(22,32,26,25);(1,5,6,4)(2,7,8,3)(17,22,23,26)(18,28,24,20)(19,31,29,27)(21,32,30,25)

Polycyclic presentation:

2^9|b^2=h,c^2=gi,e^2=hi,f^2=gi,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, 8371546);