#7713196

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×#538	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4281 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,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, 7620392);