#8209860

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	#37752	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4489 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=g,d^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8117056);