#7712600

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	#38902	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4500 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)(12,19)(16,21)(17,23)(22,24)(25,26)(27,29)(28,36)(30,32)(31,34)(33,37)(35,39)(38,40);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,22)(18,20)(19,21)(23,24)(25,27,30,34)(26,31,32,29)(28,35,37,40)(33,39,36,38);(1,3,5,4)(2,7,6,8)(10,14)(15,20)(16,21)(22,24)(26,32)(27,34)(28,37)(38,39);(9,11,13,18)(10,15,14,20)(12,17,19,23)(16,22,21,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,39)(36,37)(38,40);(1,4,5,3)(2,8,6,7)(9,10)(11,15)(12,16)(13,14)(17,22)(18,20)(19,21)(23,24)(25,28)(26,33)(27,35)(29,38)(30,37)(31,39)(32,36)(34,40);(9,12,13,19)(10,16,14,21)(11,17,18,23)(15,22,20,24)(25,29,30,31)(26,34,32,27)(28,38,37,39)(33,40,36,35)

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=hi,e^2=gh,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, 7619796);