#8889158

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	#50135	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4305 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=hi,d^2=g,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8796354);