#8850605

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

4121 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)(3,6)(4,12)(5,9)(7,11)(8,14)(10,16)(13,15)(17,18,22,26)(19,27,30,23)(20,31,29,24)(21,32,28,25)(34,36)(35,40)(38,46)(45,48);(1,3,5,11)(2,6,9,7)(4,13,14,10)(8,16,12,15)(17,19,22,30)(18,23,26,27)(20,28,29,21)(24,25,31,32)(33,34,39,36)(35,42,40,37)(38,41,46,44)(43,48,47,45);(1,2,5,9)(3,7,11,6)(4,8,14,12)(10,16,13,15)(17,20,22,29)(18,24,26,31)(19,28,30,21)(23,25,27,32)(33,35)(34,37)(36,42)(38,43)(39,40)(41,45)(44,48)(46,47);(1,2)(3,6)(4,8)(5,9)(7,11)(10,15)(12,14)(13,16)(17,20)(18,24)(19,28)(21,30)(22,29)(23,25)(26,31)(27,32)(33,36,39,34)(35,42,40,37)(38,44,46,41)(43,48,47,45);(1,3)(2,7)(4,10)(5,11)(6,9)(8,16)(12,15)(13,14)(17,21,22,28)(18,25,26,32)(19,29,30,20)(23,24,27,31)(33,37,39,42)(34,40,36,35)(38,45,46,48)(41,47,44,43);(1,4)(2,8)(3,10)(5,14)(6,15)(7,16)(9,12)(11,13)(17,18)(19,27)(20,24)(21,25)(22,26)(23,30)(28,32)(29,31)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8757801);