#8095781

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

4471 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)(9,10,14,18)(11,16,12,15)(13,17,21,24)(19,23,20,22)(25,26,31,29)(27,28,34,32)(30,37,38,33)(35,39,40,36);(1,3,5,8)(2,6,7,4)(9,11,14,12)(10,15,18,16)(13,19,21,20)(17,22,24,23)(25,27)(26,32)(28,29)(30,35)(31,34)(33,39)(36,37)(38,40);(1,3)(2,4)(5,8)(6,7)(13,21)(17,24)(19,20)(22,23)(26,29)(27,34)(33,37)(35,40);(1,4)(2,3)(5,6)(7,8)(9,12,14,11)(10,16,18,15)(13,20,21,19)(17,23,24,22)(25,28)(26,27)(29,34)(30,36)(31,32)(33,35)(37,40)(38,39);(9,13,14,21)(10,17,18,24)(11,19,12,20)(15,22,16,23)(25,29,31,26)(27,28,34,32)(30,37,38,33)(35,36,40,39);(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,30,31,38)(26,33,29,37)(27,35,34,40)(28,36,32,39)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=gi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8002977);