#8904774

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	#47993	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4197 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)(3,8)(4,14)(5,9)(6,10)(7,13)(11,12)(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)(2,7)(4,11)(5,12)(6,13)(8,10)(9,15)(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)(5,14)(11,15)(12,16)(19,27)(20,29)(23,30)(24,32)(33,35,37,43)(34,39,41,38)(36,42,45,44)(40,47,48,46);(1,4,6,9)(2,5,10,14)(3,11,13,15)(7,12,8,16)(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);(1,3)(2,8)(4,15)(5,12)(6,13)(7,10)(9,11)(14,16)(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);(1,5)(2,9)(3,12)(4,10)(6,14)(7,15)(8,11)(13,16)(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,c^2=h,d^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, 8811970);