#8455682

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4033 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2)(3,10)(4,13)(5,14)(6,8)(7,9)(11,12)(15,16)(18,24)(19,20)(26,28)(27,30)(34,38)(39,45)(40,46)(47,48);(1,3,7,6)(2,8,9,10)(4,11,14,16)(5,12,13,15)(17,18)(19,25)(20,29)(21,26)(22,27)(23,24)(28,31)(30,32)(34,38)(35,41)(40,46)(42,43);(17,19,23,20)(18,25,24,29)(21,27,31,30)(22,28,32,26)(33,34)(35,39)(36,40)(37,38)(41,45)(42,47)(43,48)(44,46);(1,4)(2,5)(3,11)(6,16)(7,14)(8,12)(9,13)(10,15)(17,20,23,19)(18,25,24,29)(21,30,31,27)(22,28,32,26)(33,35,37,41)(34,39,38,45)(36,43,44,42)(40,48,46,47);(1,5)(2,4)(3,12)(6,15)(7,13)(8,11)(9,14)(10,16)(17,21,23,31)(18,26,24,28)(19,27,20,30)(22,29,32,25);(1,6,7,3)(2,8,9,10)(4,11,14,16)(5,15,13,12)(17,22)(18,27)(19,28)(20,26)(21,29)(23,32)(24,30)(25,31)(33,36,37,44)(34,40,38,46)(35,42,41,43)(39,47,45,48)

Polycyclic presentation:

2^9|b^2=ghi,c^2=hi,d^2=h,e^2=hi,f^2=gh,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, 8362878);