#8359982

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	#55806	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4118 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gh,d^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8267178);