#8957575

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

4129 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gh,d^2=hi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,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, 8864771);