#8097028

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	#52977	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4484 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)(9,10,15,18)(11,19,22,16)(12,21,23,13)(14,17,24,20)(25,26)(27,35)(28,37)(29,31)(30,33)(32,34)(36,39)(38,40);(1,2,4,6)(3,5,8,7)(9,11,15,22)(10,16,18,19)(12,20,23,17)(13,14,21,24)(25,27,32,31)(26,29,34,35)(28,38,30,36)(33,40,37,39);(1,2)(3,5)(4,6)(7,8)(25,28,32,30)(26,33,34,37)(27,36,31,38)(29,39,35,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,13)(11,20)(14,16)(15,23)(17,22)(18,21)(19,24)(25,29)(26,27)(28,39)(30,40)(31,34)(32,35)(33,36)(37,38);(9,13)(10,12)(11,14)(15,21)(16,20)(17,19)(18,23)(22,24)(25,30,32,28)(26,33,34,37)(27,36,31,38)(29,40,35,39);(9,14)(10,17)(11,21)(12,16)(13,22)(15,24)(18,20)(19,23)(25,31,32,27)(26,29,34,35)(28,38,30,36)(33,39,37,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004224);