#8471776

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

5435 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)(3,5,7,8)(9,10,13,11)(12,15,16,14)(17,18,23,24)(19,25,20,28)(21,31,32,22)(26,30,29,27);(3,7)(5,8)(10,11)(14,15)(18,24)(21,32)(25,28)(26,29);(1,2)(3,5)(4,6)(7,8)(9,11,13,10)(12,15,16,14)(17,19,23,20)(18,25,24,28)(21,26,32,29)(22,27,31,30);(1,3,4,7)(2,5,6,8)(9,12,13,16)(10,14,11,15)(17,20,23,19)(18,25,24,28)(21,29,32,26)(22,27,31,30);(1,2)(3,5)(4,6)(7,8)(17,21)(18,22)(19,26)(20,29)(23,32)(24,31)(25,27)(28,30);(17,22)(18,21)(19,27)(20,30)(23,31)(24,32)(25,26)(28,29)

Polycyclic presentation:

2^9|a^2=gh,c^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, 8378972);