#8908183

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	#38437	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4270 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gh,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8815379);