#7743618

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	#37273	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4528 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)(11,15)(12,17)(13,18)(14,19)(16,20)(21,22)(23,24)(25,26)(27,36)(28,39)(29,30)(31,33)(32,37)(34,38)(35,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,18)(13,17)(14,20)(16,19)(21,23)(22,24)(25,27,31,37)(26,32,33,36)(28,29,38,40)(30,39,35,34);(1,3,5,4)(2,6,8,7)(11,20)(12,22)(15,16)(17,21)(26,33)(27,37)(29,40)(34,39);(1,4,5,3)(2,7,8,6)(9,11)(10,16)(12,18)(13,21)(14,20)(15,19)(17,23)(22,24)(25,28,31,38)(26,34,33,39)(27,29,37,40)(30,36,35,32);(9,12)(10,17)(11,18)(13,15)(14,22)(16,23)(19,21)(20,24)(25,29)(26,35)(27,38)(28,37)(30,33)(31,40)(32,34)(36,39);(9,13,14,23)(10,18,19,24)(11,21,20,17)(12,15,22,16)(25,30,31,35)(26,29,33,40)(27,34,37,39)(28,36,38,32)

Polycyclic presentation:

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

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, 7650814);