#8118628

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	#54015	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3907 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8025824);