#10320438

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

5315 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10227634);