#9947375

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	#48461	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4458 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=i,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9854571);