#8665386

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	#57205	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4409 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gi,d^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8572582);