#7798069

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

4356 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)(9,10,15,12)(11,17,13,16)(14,18,22,20)(19,24,21,23)(25,26)(27,36)(28,34)(29,30)(31,33)(32,37)(35,40)(38,39);(1,2)(3,7)(4,8)(5,6)(9,11,15,13)(10,16,12,17)(14,19,22,21)(18,23,20,24)(25,27)(26,32)(28,29)(30,39)(31,37)(33,36)(34,35)(38,40);(1,3,5,4)(2,7,6,8)(10,12)(16,17)(18,20)(23,24)(26,33)(27,37)(28,38)(30,35);(1,3,5,4)(2,7,6,8)(9,12,15,10)(11,17,13,16)(14,20,22,18)(19,24,21,23)(25,28)(26,34)(27,29)(30,36)(31,38)(32,35)(33,39)(37,40);(1,4,5,3)(2,8,6,7)(9,13,15,11)(10,17,12,16)(14,21,22,19)(18,24,20,23)(25,29)(26,35)(27,38)(28,37)(30,33)(31,40)(32,39)(34,36);(9,14,15,22)(10,18,12,20)(11,19,13,21)(16,23,17,24)(25,30,31,35)(26,29,33,40)(27,39,37,34)(28,36,38,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gh,d^2=gi,e^2=gi,f^2=h,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, 7705265);