#8436756

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	#40842	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4407 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=gi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8343952);