#9839274

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	#45643	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4247 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)(3,4)(5,7)(6,8)(9,10,14,17)(11,15,20,23)(12,13,21,16)(18,19,24,22)(25,26)(27,31)(28,32)(29,37)(30,34)(33,39)(35,36)(38,40);(1,3,6,7)(2,5,8,4)(9,11,14,20)(10,15,17,23)(12,18,21,24)(13,19,16,22)(25,27,30,32)(26,31,34,28)(29,35,39,40)(33,38,37,36);(9,12,14,21)(10,16,17,13)(11,18,20,24)(15,22,23,19)(25,26,30,34)(27,28,32,31)(29,33,39,37)(35,36,40,38);(1,4,6,5)(2,7,8,3)(9,12)(10,13)(11,18)(14,21)(15,19)(16,17)(20,24)(22,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,40)(37,39);(1,5,6,4)(2,7,8,3)(9,11)(10,15)(12,18)(13,19)(14,20)(16,22)(17,23)(21,24)(25,28,30,31)(26,32,34,27)(29,38,39,36)(33,35,37,40);(1,2)(3,5)(4,7)(6,8)(9,13)(10,12)(11,19)(14,16)(15,18)(17,21)(20,22)(23,24)(25,29)(26,33)(27,35)(28,36)(30,39)(31,38)(32,40)(34,37)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,d^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,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, 9746470);