#7720356

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4165 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

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, 7627552);