#8094842

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	#53134	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4198 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)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26)(27,34)(28,29)(30,31)(32,37)(33,38)(35,39)(36,40);(1,2,5,3)(4,6,8,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,20)(23,24)(26,31)(28,38)(32,34)(35,40);(2,3)(6,7)(11,17)(14,21)(18,23)(22,24)(25,27,30,37)(26,32,31,34)(28,35,38,40)(29,36,33,39);(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,20)(23,24)(25,28)(26,33)(27,35)(29,31)(30,38)(32,39)(34,36)(37,40);(1,3,5,2)(4,7,8,6)(9,11,13,17)(10,14,16,21)(12,18,20,23)(15,22,19,24);(1,4)(2,6)(3,7)(5,8)(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(25,29,30,33)(26,28,31,38)(27,36,37,39)(32,35,34,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,e^2=ghi,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, 8002038);