#8094886

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	#45720	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4206 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=gh,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

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, 8002082);