#8912257

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

4206 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)(3,11)(4,14)(5,8)(6,9)(7,10)(12,13)(15,16)(17,18,22,25)(19,23,26,20)(21,24,30,32)(27,29,31,28)(33,34,38,39)(35,36,41,40)(37,45,47,42)(43,48,46,44);(1,3,7,5)(2,8,10,11)(4,12,9,15)(6,13,14,16)(17,19)(18,23)(20,25)(21,27)(22,26)(24,29)(28,32)(30,31)(34,39)(36,40)(42,45)(44,48);(4,9)(6,14)(12,15)(13,16)(19,26)(20,23)(27,31)(28,29)(33,35,38,41)(34,40,39,36)(37,46,47,43)(42,44,45,48);(1,4,7,9)(2,6,10,14)(3,12,5,15)(8,13,11,16)(17,20,22,23)(18,19,25,26)(21,29,30,28)(24,31,32,27)(33,35)(34,40)(36,39)(37,43)(38,41)(42,48)(44,45)(46,47);(1,5,7,3)(2,8,10,11)(4,12,9,15)(6,16,14,13)(17,21)(18,24)(19,27)(20,28)(22,30)(23,29)(25,32)(26,31)(33,36,38,40)(34,41,39,35)(37,44,47,48)(42,46,45,43);(1,6)(2,9)(3,13)(4,10)(5,16)(7,14)(8,15)(11,12)(17,19)(18,23)(20,25)(21,27)(22,26)(24,29)(28,32)(30,31)(33,37,38,47)(34,42,39,45)(35,43,41,46)(36,44,40,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8819453);