#8276609

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	#40661	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4214 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)(3,6)(4,13)(5,16)(7,9)(8,12)(10,15)(11,14)(17,18,22,25)(19,23,26,30)(20,21,24,29)(27,28,31,32)(33,34)(35,40)(36,39)(37,41)(38,42)(43,47)(44,46)(45,48);(1,3,7,12)(2,8,9,6)(4,14,15,5)(10,11,13,16)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32)(33,35,38,36)(34,39,42,40)(37,43,45,44)(41,46,48,47);(2,9)(6,8)(10,13)(11,16)(19,26)(23,30)(27,31)(28,32)(33,35,38,36)(34,40,42,39)(37,43,45,44)(41,47,48,46);(1,4,7,15)(2,10,9,13)(3,5,12,14)(6,11,8,16)(17,20)(18,21)(19,27)(22,24)(23,28)(25,29)(26,31)(30,32)(33,36,38,35)(34,40,42,39)(37,43,45,44)(41,46,48,47);(1,5)(2,11)(3,4)(6,10)(7,14)(8,13)(9,16)(12,15)(17,21)(18,24)(19,28)(20,25)(22,29)(23,31)(26,32)(27,30)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,6)(2,3)(4,11)(5,10)(7,8)(9,12)(13,14)(15,16)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8183805);