#8424739

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	#42581	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4485 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8331935);