#8324123

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

5606 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8231319);