#8110708

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

5040 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8017904);