#8461543

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

4793 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=gh,e^2=ghi,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, 8368739);