#8109944

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	#2856	C₂²
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) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8017140);