#8702255

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

5226 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8609451);