#7710148

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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4318 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=hi,e^2=h,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, 7617344);