#8757166

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

4051 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=hi,d^2=g,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

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, 8664362);