#8508934

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

4922 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)(3,7)(4,5)(6,8)(9,10,12,13)(11,15,16,14)(18,24)(19,20)(26,29)(27,28);(1,3)(2,6)(4,7)(5,8)(10,13)(14,15)(17,18)(19,25)(20,22)(21,29)(23,24)(26,31)(27,30)(28,32);(9,10)(11,14)(12,13)(15,16)(17,19,23,20)(18,25,24,22)(21,27,31,28)(26,32,29,30);(1,2,5,4)(3,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,20,23,19)(18,25,24,22)(21,28,31,27)(26,32,29,30);(1,4,5,2)(3,6,8,7)(9,11,12,16)(10,14,13,15)(17,21)(18,26)(19,27)(20,28)(22,30)(23,31)(24,29)(25,32);(1,3)(2,7)(4,6)(5,8)(17,22,23,25)(18,20,24,19)(21,30,31,32)(26,28,29,27)

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=gi,e^2=ghi,f^2=hi,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, 8416130);