#10320439

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

4899 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)(18,23)(19,24)(25,31)(26,32);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,13)(14,16)(17,18,22,23)(19,20,24,28)(21,25,30,31)(26,27,32,29);(1,2,5,3)(4,6,8,7)(9,10,14,12)(11,15,16,13)(17,19,22,24)(18,20,23,28)(21,26,30,32)(25,27,31,29);(2,3)(6,7)(10,12)(11,16)(17,18)(19,20)(21,25)(22,23)(24,28)(26,27)(29,32)(30,31);(1,3,5,2)(4,6,8,7)(9,12,14,10)(11,15,16,13)(17,20)(18,24)(19,23)(21,29)(22,28)(25,26)(27,30)(31,32);(1,4,5,8)(2,6,3,7)(9,13,14,15)(10,11,12,16)(17,21,22,30)(18,25,23,31)(19,26,24,32)(20,27,28,29)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,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, 10227635);