#8912293

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

4229 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=ghi,e^2=g,f^2=gh,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, 8819489);