#10320424

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	#43481	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4315 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)(9,10,14,11)(12,15,19,17)(13,18,21,16)(20,23,24,22)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,39)(38,40);(2,6)(3,7)(12,19)(15,17)(20,24)(22,23)(25,27,31,32)(26,28,34,29)(30,35,38,39)(33,36,40,37);(1,3)(2,4)(5,7)(6,8)(10,11)(15,17)(16,18)(22,23)(25,28,31,29)(26,32,34,27)(30,36,38,37)(33,39,40,35);(1,4)(2,3)(5,8)(6,7)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,27)(26,28)(29,34)(30,35)(31,32)(33,36)(37,40)(38,39);(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,29,31,28)(26,32,34,27)(30,36,38,37)(33,35,40,39);(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,30,31,38)(26,33,34,40)(27,35,32,39)(28,36,29,37)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gi,d^2=ghi,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, 10227620);