#8973360

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

4065 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,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, 8880556);