#8892808

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	#47694	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4340 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8800004);