#8839159

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

3948 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)(3,7)(4,8)(5,14)(6,10)(9,15)(11,12)(13,16)(17,18,23,28)(19,26,21,24)(20,25,31,29)(22,30,32,27)(33,34,39,37)(35,42,40,36)(38,45,46,41)(43,44,47,48);(1,3)(2,7)(4,5)(6,11)(8,14)(9,16)(10,12)(13,15)(19,21)(22,32)(24,26)(27,30)(34,37)(35,40)(41,45)(43,47);(3,11)(4,13)(7,12)(8,16)(17,19,23,21)(18,24,28,26)(20,30,31,27)(22,29,32,25)(33,35)(34,36)(37,42)(38,43)(39,40)(41,44)(45,48)(46,47);(1,4,6,13)(2,8,10,16)(3,5,11,15)(7,14,12,9)(17,20)(18,25)(19,27)(21,30)(22,26)(23,31)(24,32)(28,29)(33,36,39,42)(34,40,37,35)(38,44,46,48)(41,47,45,43);(1,5)(2,9)(3,4)(6,15)(7,16)(8,12)(10,14)(11,13)(17,21,23,19)(18,26,28,24)(20,27,31,30)(22,25,32,29)(33,37,39,34)(35,42,40,36)(38,45,46,41)(43,48,47,44);(1,2)(3,12)(4,8)(5,9)(6,10)(7,11)(13,16)(14,15)(17,22,23,32)(18,27,28,30)(19,29,21,25)(20,26,31,24)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=ghi,e^2=h,f^2=gh,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, 8746355);