#10020329

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

4537 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=ghi,d^2=ghi,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dghi,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9927525);