#8186994

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	#34963	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4645 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)(3,7)(4,5)(6,8)(9,10,14,15)(11,17,19,13)(12,20,22,16)(18,21,24,23)(25,26,30,33)(27,31,34,28)(29,32,38,40)(35,39,37,36);(2,5)(6,7)(9,11)(10,13)(12,18)(14,19)(15,17)(16,21)(20,23)(22,24)(27,34)(28,31)(35,37)(36,39);(1,2)(3,6)(4,5)(7,8)(10,15)(13,17)(16,20)(21,23)(25,27)(26,31)(28,33)(29,37)(30,34)(32,36)(35,38)(39,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,13)(12,18)(14,19)(15,17)(16,21)(20,23)(22,24)(25,28,30,31)(26,27,33,34)(29,36,38,39)(32,35,40,37);(9,12,14,22)(10,16,15,20)(11,18,19,24)(13,21,17,23);(9,13)(10,11)(12,21)(14,17)(15,19)(16,18)(20,24)(22,23)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,40)(34,37)

Polycyclic presentation:

2^9|a^2=gi,d^2=ghi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

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, 8094190);