#7724568

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	#47090	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4237 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,d^2=i,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, 7631764);