#8094857

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	#53106	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4325 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)(11,16)(12,13)(14,15)(17,21)(18,22)(19,20)(23,24)(25,26,31,36)(27,29,34,32)(28,35,30,33)(37,40,38,39);(1,2,5,3)(4,6,8,7)(11,18)(16,22)(19,23)(20,24)(25,27)(26,32)(28,37)(29,36)(30,38)(31,34)(33,40)(35,39);(2,3)(6,7)(10,15)(12,17)(16,22)(19,23)(25,27)(26,29)(28,37)(30,38)(31,34)(32,36)(33,39)(35,40);(9,11,14,18)(10,16,15,22)(12,19,17,23)(13,20,21,24)(25,28,31,30)(26,33,36,35)(27,37,34,38)(29,39,32,40);(1,3,5,2)(4,7,8,6)(9,12,14,17)(10,13,15,21)(11,19,18,23)(16,20,22,24)(25,29)(26,34)(27,36)(28,39)(30,40)(31,32)(33,38)(35,37);(1,4)(2,6)(3,7)(5,8)(9,13)(10,17)(11,20)(12,15)(14,21)(16,23)(18,24)(19,22)(25,30,31,28)(26,35,36,33)(27,38,34,37)(29,40,32,39)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=gh,e^2=hi,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, 8002053);