#7895109

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

4385 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=hi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7802305);