#10320332

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	#48930	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4216 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)(9,10,15,13)(11,16,19,12)(14,21,23,17)(18,20,22,24)(25,26,30,31)(27,35,37,32)(28,36,38,33)(29,34,39,40);(2,6)(3,7)(9,11,15,19)(10,16,13,12)(14,18,23,22)(17,24,21,20)(26,31)(28,38)(32,35)(34,40);(1,3)(2,4)(5,7)(6,8)(9,12,15,16)(10,11,13,19)(14,20,23,24)(17,18,21,22)(25,27)(26,32)(28,34)(29,36)(30,37)(31,35)(33,39)(38,40);(1,4)(2,3)(5,8)(6,7)(9,13,15,10)(11,16,19,12)(14,21,23,17)(18,24,22,20)(25,28,30,38)(26,33,31,36)(27,34,37,40)(29,32,39,35);(9,11)(10,16)(12,13)(14,22)(15,19)(17,20)(18,23)(21,24)(25,29,30,39)(26,34,31,40)(27,36,37,33)(28,35,38,32);(9,14)(10,17)(11,18)(12,20)(13,21)(15,23)(16,24)(19,22)(25,27)(26,35)(28,34)(29,33)(30,37)(31,32)(36,39)(38,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=ghi,d^2=gi,e^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, 10227528);