#8093445

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

4139 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)(3,8)(4,9)(5,12)(6,7)(10,15)(11,14)(13,16)(20,28)(24,29)(26,32)(30,31)(33,34)(35,41)(36,40)(37,45)(38,48)(39,42)(43,46)(44,47);(3,11)(4,13)(8,14)(9,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35,39,45)(34,37,42,41)(36,43,38,44)(40,47,48,46);(2,7)(3,11)(9,16)(10,12)(18,22)(23,25)(24,29)(30,31)(33,36,39,38)(34,40,42,48)(35,43,45,44)(37,47,41,46);(1,3)(2,8)(4,12)(5,9)(6,11)(7,14)(10,13)(15,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,37,39,41)(34,35,42,45)(36,46,38,47)(40,44,48,43);(1,4,6,13)(2,9,7,16)(3,12,11,10)(5,14,15,8)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,35)(34,41)(36,43)(37,42)(38,44)(39,45)(40,46)(47,48);(1,5)(2,10)(3,9)(4,14)(6,15)(7,12)(8,13)(11,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,38,39,36)(34,40,42,48)(35,44,45,43)(37,47,41,46)

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi,d^c=dghi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000641);