#7754834

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	#43034	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4406 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:

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

Polycyclic presentation:

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

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, 7662030);