#8110508

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	#46225	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4520 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)(11,17)(12,18)(13,16)(14,15)(19,23)(20,24)(21,22)(25,26,30,34)(27,28,32,31)(29,37,38,33)(35,39,40,36);(2,4)(7,8)(10,15)(11,18)(16,22)(19,24)(27,32)(28,31)(29,38)(33,37);(1,3,6,5)(2,7,4,8)(9,11)(10,12)(13,19)(14,18)(15,17)(16,20)(21,24)(22,23)(25,27,30,32)(26,31,34,28)(29,35,38,40)(33,39,37,36);(1,2,6,4)(3,7,5,8)(9,11)(10,12)(13,19)(14,18)(15,17)(16,20)(21,24)(22,23)(25,28,30,31)(26,32,34,27)(29,36,38,39)(33,40,37,35);(1,4,6,2)(3,8,5,7)(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,23)(22,24)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,40)(34,37);(1,5,6,3)(2,8,4,7)(9,13,14,21)(10,16,15,22)(11,19,18,24)(12,20,17,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,39)(36,40)(37,38)

Polycyclic presentation:

2^9|a^2=gh,c^2=gi,d^2=gi,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8017704);