#8295644

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

4012 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,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, 8202840);