#8120009

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	#53476	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4207 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)(9,10,15,16)(11,13,12,17)(14,18,22,23)(19,21,20,24)(25,26,31,29)(27,34,30,32)(28,33,37,38)(35,40,36,39);(1,2,5,3)(4,6,8,7)(10,16)(11,12)(18,23)(19,20)(25,27,31,30)(26,32,29,34)(28,35,37,36)(33,39,38,40);(2,3)(6,7)(9,11,15,12)(10,17,16,13)(14,19,22,20)(18,24,23,21)(25,28)(26,33)(27,35)(29,38)(30,36)(31,37)(32,39)(34,40);(9,12,15,11)(10,13,16,17)(14,20,22,19)(18,21,23,24)(25,27,31,30)(26,32,29,34)(28,35,37,36)(33,39,38,40);(1,3,5,2)(4,7,8,6)(9,13)(10,11)(12,16)(14,21)(15,17)(18,19)(20,23)(22,24)(25,29,31,26)(27,34,30,32)(28,33,37,38)(35,39,36,40);(1,4)(2,6)(3,7)(5,8)(9,14,15,22)(10,18,16,23)(11,19,12,20)(13,21,17,24)(25,30,31,27)(26,34,29,32)(28,36,37,35)(33,40,38,39)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,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^c=egi

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, 8027205);