#8942629

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	#56589	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4274 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=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, 8849825);