#8295588

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	#38077	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4444 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:

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

Polycyclic presentation:

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

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, 8202784);