#7720494

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	#37276	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4358 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:

(2,5)(6,8)(9,10,15,12)(11,17,13,16)(14,20,22,18)(19,23,21,24)(25,26,31,28)(27,35,36,32)(29,33,39,38)(30,37,40,34);(1,2,4,5)(3,6,7,8)(9,11)(10,16)(12,17)(13,15)(14,19)(18,23)(20,24)(21,22)(25,27,31,36)(26,32,28,35)(29,34,39,37)(30,33,40,38);(10,12)(11,13)(18,20)(19,21)(27,36)(29,39)(32,35)(33,38);(1,3,4,7)(2,6,5,8)(9,12,15,10)(11,17,13,16)(14,20,22,18)(19,24,21,23)(25,28,31,26)(27,35,36,32)(29,38,39,33)(30,37,40,34);(9,13,15,11)(10,17,12,16)(14,21,22,19)(18,24,20,23)(25,29)(26,33)(27,37)(28,38)(30,32)(31,39)(34,36)(35,40);(9,14,15,22)(10,18,12,20)(11,19,13,21)(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=i,b^2=gh,d^2=g,e^2=h,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, 7627690);