#8471276

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

4989 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,e^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, 8378472);