#8924641

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

4067 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=hi,d^2=hi,e^2=i,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, 8831837);