#8658125

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

5151 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8565321);