#7798081

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

4502 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=h,d^2=gi,e^2=i,f^2=gh,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, 7705277);