#8912212

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

4091 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gh,d^2=ghi,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, 8819408);