#8757623

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	#51930	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4322 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=ghi,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8664819);