#7714996

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

4435 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=gh,d^2=hi,e^2=gi,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=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, 7622192);