#8471712

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

5365 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8378908);