#7918388

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

4593 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=g,d^2=ghi,e^2=gi,f^2=ghi,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, 7825584);