#8285300

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	#39392	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4353 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)(3,8)(4,6)(5,7)(9,10,15,18)(11,16,19,14)(12,13,17,23)(20,21,24,22)(27,28)(31,34)(35,36)(38,39);(1,2,6,4)(3,8,5,7)(9,11,15,19)(10,16,18,14)(12,20,17,24)(13,21,23,22)(25,26,30,33)(27,34,28,31)(29,32,37,40)(35,39,36,38);(1,2,6,4)(3,7,5,8)(11,19)(14,16)(20,24)(21,22)(25,27,30,28)(26,31,33,34)(29,36,37,35)(32,39,40,38);(1,3,6,5)(2,7,4,8)(9,12)(10,13)(11,20)(14,22)(15,17)(16,21)(18,23)(19,24)(25,26)(27,31)(28,34)(29,32)(30,33)(35,38)(36,39)(37,40);(1,4,6,2)(3,7,5,8)(9,13)(10,17)(11,21)(12,18)(14,20)(15,23)(16,24)(19,22)(25,28,30,27)(26,31,33,34)(29,36,37,35)(32,38,40,39);(1,5,6,3)(2,7,4,8)(9,14)(10,11)(12,22)(13,20)(15,16)(17,21)(18,19)(23,24)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,40)(34,39)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=h,d^2=g,e^2=h,f^2=g,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, 8192496);