#8192540

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	#49637	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4071 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)(3,7)(4,11)(5,6)(8,12)(9,15)(10,13)(14,16)(34,36)(35,41)(40,44)(43,47);(1,2)(3,7)(4,8)(5,6)(9,14)(10,13)(11,12)(15,16)(17,18,22,23)(19,25,27,21)(20,24,29,30)(26,31,32,28)(33,34,38,36)(35,39,41,42)(37,40,45,44)(43,46,47,48);(2,6)(7,13)(8,11)(14,15)(18,23)(21,25)(24,30)(28,31)(33,34,38,36)(35,39,41,42)(37,44,45,40)(43,48,47,46);(1,3,5,10)(2,7,6,13)(4,9,12,16)(8,14,11,15)(17,19)(18,21)(20,26)(22,27)(23,25)(24,28)(29,32)(30,31)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,4)(2,8)(3,9)(5,12)(6,11)(7,14)(10,16)(13,15)(17,20,22,29)(18,24,23,30)(19,26,27,32)(21,28,25,31)(33,36,38,34)(35,42,41,39)(37,44,45,40)(43,48,47,46);(1,2)(3,7)(4,8)(5,6)(9,14)(10,13)(11,12)(15,16)(17,21)(18,19)(20,28)(22,25)(23,27)(24,26)(29,31)(30,32)(33,37,38,45)(34,40,36,44)(35,43,41,47)(39,46,42,48)

Polycyclic presentation:

2^9|b^2=hi,c^2=g,d^2=gh,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8099736);