#8155054

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4228 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8062250);