#8436865

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	#57954	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4384 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8344061);