#7710150

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	#2847	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4632 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)(3,5,7,8)(12,19)(16,21)(17,23)(22,24)(25,26)(27,31)(28,37)(29,38)(30,32)(33,36)(34,39)(35,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,22)(18,20)(19,21)(23,24)(25,27,30,35)(26,31,32,40)(28,29,36,39)(33,34,37,38);(1,3,4,7)(2,5,6,8)(10,14)(15,20)(16,21)(22,24)(26,32)(27,35)(28,36)(34,38);(9,11,13,18)(10,15,14,20)(12,17,19,23)(16,22,21,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,40)(36,37)(38,39);(9,10)(11,15)(12,16)(13,14)(17,22)(18,20)(19,21)(23,24)(25,28,30,36)(26,33,32,37)(27,29,35,39)(31,34,40,38);(9,12)(10,16)(11,17)(13,19)(14,21)(15,22)(18,23)(20,24)(25,29)(26,34)(27,36)(28,35)(30,39)(31,37)(32,38)(33,40)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,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, 7617346);