#7710922

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	#45480	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4464 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)(11,19)(12,18)(13,16)(14,20)(15,17)(21,22)(23,24)(25,26)(27,31)(28,37)(29,38)(30,32)(33,36)(34,39)(35,40);(1,2)(3,7)(4,8)(5,6)(9,11,15,13)(10,16,17,19)(12,21,20,23)(14,22,18,24)(25,27,30,35)(26,31,32,40)(28,29,36,39)(33,34,37,38);(10,17)(12,20)(16,19)(21,23)(26,32)(27,35)(28,36)(34,38);(1,3,5,4)(2,7,6,8)(9,12,15,20)(10,18,17,14)(11,21,13,23)(16,24,19,22)(25,26)(27,31)(28,33)(29,34)(30,32)(35,40)(36,37)(38,39);(1,3,5,4)(2,7,6,8)(9,13,15,11)(10,19,17,16)(12,23,20,21)(14,24,18,22)(25,28,30,36)(26,33,32,37)(27,29,35,39)(31,34,40,38);(1,4,5,3)(2,8,6,7)(9,14,15,18)(10,20,17,12)(11,22,13,24)(16,23,19,21)(25,29)(26,34)(27,36)(28,35)(30,39)(31,37)(32,38)(33,40)

Polycyclic presentation:

2^9|b^2=h,d^2=g,e^2=gi,f^2=g,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, 7618118);