#7710007

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

4524 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=i,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, 7617203);