#7720508

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	#37278	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4557 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,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, 7627704);