#8904786

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
prime radical	#45591	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4089 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)(17,18,22,26)(19,23,27,30)(20,24,29,32)(21,25,31,28)(33,34,37,41)(35,38,43,39)(36,40,45,48)(42,46,44,47);(1,3,7,5)(2,8,10,11)(4,12,9,15)(6,13,14,16)(17,19)(18,23)(20,28)(21,24)(22,27)(25,29)(26,30)(31,32)(33,35)(34,38)(36,44)(37,43)(39,41)(40,47)(42,45)(46,48);(4,9)(6,14)(12,15)(13,16)(19,27)(20,29)(23,30)(24,32)(33,35,37,43)(34,39,41,38)(36,42,45,44)(40,47,48,46);(1,4,7,9)(2,6,10,14)(3,12,5,15)(8,13,11,16)(17,20)(18,24)(19,28)(21,23)(22,29)(25,27)(26,32)(30,31)(33,35)(34,38)(36,44)(37,43)(39,41)(40,47)(42,45)(46,48);(1,5,7,3)(2,8,10,11)(4,12,9,15)(6,16,14,13)(17,21)(18,25)(19,24)(20,30)(22,31)(23,29)(26,28)(27,32)(33,36)(34,40)(35,42)(37,45)(38,46)(39,47)(41,48)(43,44);(1,6)(2,9)(3,13)(4,10)(5,16)(7,14)(8,15)(11,12)(17,19)(18,23)(20,28)(21,24)(22,27)(25,29)(26,30)(31,32)(33,34)(35,38)(36,40)(37,41)(39,43)(42,46)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=g,e^2=g,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, 8811982);