#7799557

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

4528 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=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, 7706753);