#8596436

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	#45445	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4389 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=g,d^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8503632);