#8174320

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

4228 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=i,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8081516);