#8946225

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	#43194	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4014 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=ghi,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8853421);