#7811052

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
prime radical	#45135	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4248 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:

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

Polycyclic presentation:

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

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, 7718248);