#8751142

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	#53115	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4230 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:

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

Polycyclic presentation:

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

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, 8658338);