#8109992

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	#39548	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4472 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gi,d^2=g,e^2=ghi,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, 8017188);