#8176302

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
prime radical	#58991	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5669 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)(3,7)(4,5)(6,8)(17,18,22,23)(19,25,27,21)(20,28,30,24)(26,29,32,31);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,13)(14,16)(17,19)(18,21)(20,26)(22,27)(23,25)(24,29)(28,31)(30,32);(2,5)(6,7)(10,13)(14,15)(18,23)(21,25)(24,28)(29,31);(9,11)(10,14)(12,16)(13,15)(17,20,22,30)(18,24,23,28)(19,26,27,32)(21,29,25,31);(1,3)(2,6)(4,8)(5,7)(17,19)(18,21)(20,26)(22,27)(23,25)(24,29)(28,31)(30,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,21)(18,19)(20,29)(22,25)(23,27)(24,26)(28,32)(30,31)

Polycyclic presentation:

2^9|a^2=g,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8083498);