#8425390

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

5078 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=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, 8332586);