#10320407

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
prime radical	#44158	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4357 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2,5,6)(3,4,7,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26,32,36)(27,33,31,35)(28,30,29,34)(37,40,38,39);(2,6)(3,7)(11,17)(15,21)(18,23)(22,24)(25,27,32,31)(26,33,36,35)(28,38,29,37)(30,39,34,40);(1,3)(2,4)(5,7)(6,8)(10,14)(15,21)(16,19)(22,24)(25,28,32,29)(26,34,36,30)(27,37,31,38)(33,39,35,40);(1,4)(2,3)(5,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,29,32,28)(26,34,36,30)(27,38,31,37)(33,39,35,40);(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24)(25,30,32,34)(26,29,36,28)(27,39,31,40)(33,37,35,38);(9,12,13,20)(10,16,14,19)(11,18,17,23)(15,22,21,24)(25,31,32,27)(26,35,36,33)(28,38,29,37)(30,39,34,40)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gi,d^2=gi,e^2=h,f^2=h,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, 10227603);