#8977148

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4090 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)(3,7)(4,12)(5,6)(8,13)(9,10)(11,15)(14,16)(18,23)(19,25)(24,30)(26,31)(33,34)(35,42)(36,44)(37,41)(38,39)(40,45)(43,47)(46,48);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,18)(19,20)(21,24)(22,23)(25,27)(26,28)(29,30)(31,32)(33,35,39,41)(34,37,38,42)(36,43,45,48)(40,47,44,46);(2,6)(3,10)(4,13)(14,15)(19,25)(20,27)(21,29)(24,30)(34,38)(36,45)(37,42)(43,48);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,19,22,25)(18,20,23,27)(21,26,29,31)(24,28,30,32)(33,36,39,45)(34,40,38,44)(35,43,41,48)(37,47,42,46);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,20)(18,19)(21,28)(22,27)(23,25)(24,26)(29,32)(30,31)(33,37,39,42)(34,41,38,35)(36,46,45,47)(40,43,44,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,21,22,29)(18,24,23,30)(19,26,25,31)(20,28,27,32)(33,38,39,34)(35,42,41,37)(36,44,45,40)(43,46,48,47)

Polycyclic presentation:

2^9|b^2=hi,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=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, 8884344);