#8730763

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

5054 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)(6,7)(9,10,13,11)(12,15,16,14)(17,18,22,26)(19,23,27,24)(20,21,25,31)(28,29,32,30);(1,2)(3,7)(4,5)(6,8)(10,11)(14,15)(17,19)(18,23)(20,28)(21,29)(22,27)(24,26)(25,32)(30,31);(1,2)(3,7)(4,5)(6,8)(9,11,13,10)(12,15,16,14)(19,27)(23,24)(28,32)(29,30);(1,3,4,8)(2,6,5,7)(9,12,13,16)(10,14,11,15)(17,19)(18,24)(20,28)(21,30)(22,27)(23,26)(25,32)(29,31);(1,2)(3,6)(4,5)(7,8)(17,20)(18,21)(19,28)(22,25)(23,29)(24,30)(26,31)(27,32);(1,3)(2,7)(4,8)(5,6)(17,21)(18,25)(19,29)(20,26)(22,31)(23,32)(24,28)(27,30)

Polycyclic presentation:

2^9|a^2=i,c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8637959);