#8757159

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	#45978	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4184 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=hi,d^2=g,e^2=ghi,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, 8664355);