#7724562

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	#36758	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4490 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=gi,e^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, 7631758);