#7712422

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

4301 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7619618);