#8093217

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

4130 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(18,22)(23,28)(24,29)(30,32)(33,34)(35,41)(36,37)(38,47)(39,40)(42,46)(43,44)(45,48);(2,6)(3,10)(8,12)(11,16)(19,25)(20,27)(23,28)(24,29)(34,40)(37,42)(41,48)(43,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,35,39,45)(34,41,40,48)(36,44,46,38)(37,43,42,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,36)(34,42)(35,44)(37,40)(38,45)(39,46)(41,47)(43,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,37,39,42)(34,36,40,46)(35,43,45,47)(38,41,44,48);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,38,39,44)(34,43,40,47)(35,36,45,46)(37,41,42,48)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000413);