#8471271

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

4980 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=g,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, 8378467);