#8912222

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

4068 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gh,d^2=hi,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, 8819418);