#8276605

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	#2617	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4120 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,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, 8183801);