#8012048

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

4476 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7919244);