#7710019

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	#59043	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4418 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7617215);