#8720500

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	#41877	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4302 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)(9,10,15,12)(11,16,17,13)(14,18,23,20)(19,22,24,21)(25,26)(27,29)(28,36)(30,31)(32,34)(33,38)(35,39)(37,40);(2,3)(6,7)(9,11,15,17)(10,16,12,13)(14,19,23,24)(18,22,20,21)(26,31)(27,34)(33,36)(35,40);(1,3,5,2)(4,7,8,6)(10,12)(11,17)(18,20)(19,24)(25,27)(26,32)(28,35)(29,31)(30,34)(33,39)(36,37)(38,40);(1,4)(2,6)(3,7)(5,8)(9,12,15,10)(11,16,17,13)(14,18,23,20)(19,21,24,22)(25,27)(26,32)(28,35)(29,31)(30,34)(33,39)(36,37)(38,40);(9,13,15,16)(10,17,12,11)(14,22,23,21)(18,19,20,24)(25,28,30,38)(26,33,31,36)(27,35,34,40)(29,37,32,39);(9,14,15,23)(10,18,12,20)(11,19,17,24)(13,21,16,22)(25,29,30,32)(26,27,31,34)(28,37,38,39)(33,35,36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8627696);