#7918184

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

4280 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)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22)(25,26)(27,38)(28,35)(29,40)(30,34)(31,36)(32,37)(33,39);(1,2)(3,7)(4,8)(5,6)(25,27)(26,33)(28,29)(30,36)(31,35)(32,39)(34,40)(37,38);(1,3,5,4)(2,7,6,8)(10,11)(15,17)(16,18)(22,23)(25,28,32,30)(26,34,37,35)(27,36,39,29)(31,38,40,33);(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,29,32,36)(26,31,37,40)(27,28,39,30)(33,35,38,34);(1,3,5,4)(2,7,6,8)(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,23)(21,24)(25,30,32,28)(26,35,37,34)(27,36,39,29)(31,38,40,33);(1,4,5,3)(2,8,6,7)(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,31)(26,36)(27,35)(28,38)(29,37)(30,33)(32,40)(34,39)

Polycyclic presentation:

2^9|a^2=hi,c^2=g,d^2=h,e^2=g,f^2=gh,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, 7825380);