#7724579

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	#40403	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4235 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)(25,26,31,28)(27,35,37,32)(29,38,40,33)(30,34,36,39);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,19)(13,14)(16,20)(17,23)(18,21)(22,24)(25,27,31,37)(26,32,28,35)(29,36,40,30)(33,39,38,34);(1,3,5,4)(2,7,6,8)(10,14)(15,21)(16,19)(22,23)(26,28)(32,35)(33,38)(34,39);(1,3,5,4)(2,7,6,8)(9,11,13,18)(10,15,14,21)(12,17,20,24)(16,22,19,23)(25,28,31,26)(27,35,37,32)(29,38,40,33)(30,39,36,34);(9,10)(11,15)(12,16)(13,14)(17,22)(18,21)(19,20)(23,24)(25,29)(26,33)(27,36)(28,38)(30,37)(31,40)(32,39)(34,35);(1,4,5,3)(2,8,6,7)(9,12)(10,16)(11,17)(13,20)(14,19)(15,22)(18,24)(21,23)(25,30,31,36)(26,34,28,39)(27,29,37,40)(32,33,35,38)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7631775);