#8295541

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	#2656	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4026 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=gi,e^2=h,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, 8202737);