#7848801

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

4538 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7755997);