#7947485

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
prime radical	#45507	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4304 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=ghi,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, 7854681);