#7986353

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	#42942	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4573 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)(11,17,13,16)(14,18,22,20)(19,24,21,23)(25,26)(27,30)(28,36)(29,31)(32,37)(33,35)(34,40)(38,39);(1,2)(3,6)(4,7)(5,8)(9,11,15,13)(10,16,12,17)(14,19,22,21)(18,23,20,24)(25,27)(26,30)(28,34)(29,35)(31,33)(32,38)(36,40)(37,39);(1,3,5,4)(2,6,8,7)(10,12)(16,17)(18,20)(23,24)(26,29)(27,33)(28,37)(38,40);(9,12,15,10)(11,17,13,16)(14,20,22,18)(19,24,21,23)(25,28,31,37)(26,32,29,36)(27,34,33,39)(30,38,35,40);(9,13,15,11)(10,17,12,16)(14,21,22,19)(18,24,20,23)(25,29,31,26)(27,35,33,30)(28,36,37,32)(34,40,39,38);(1,4,5,3)(2,7,8,6)(9,14,15,22)(10,18,12,20)(11,19,13,21)(16,23,17,24)(25,30,31,35)(26,27,29,33)(28,38,37,40)(32,34,36,39)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7893549);