#8136847

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

4239 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)(8,15)(10,16)(13,14)(17,18)(19,29)(20,26)(21,22)(23,32)(24,28)(25,31)(27,30)(33,34)(35,40)(36,37)(38,43)(39,42)(41,48)(44,47)(45,46);(1,2)(3,7)(4,8)(5,9)(6,11)(10,14)(12,15)(13,16)(17,19,24,31)(18,25,28,29)(20,30,21,23)(22,27,26,32)(33,35,39,38)(34,40,42,43)(36,46,41,44)(37,45,48,47);(1,3)(2,6)(4,10)(5,11)(7,9)(8,13)(12,16)(14,15)(17,20,24,21)(18,26,28,22)(19,30,31,23)(25,32,29,27)(33,34,39,42)(35,43,38,40)(36,37,41,48)(44,45,46,47);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,21,24,20)(18,26,28,22)(19,23,31,30)(25,32,29,27)(33,36)(34,37)(35,44)(38,46)(39,41)(40,47)(42,48)(43,45);(1,2)(3,6)(4,8)(5,9)(7,11)(10,13)(12,15)(14,16)(17,22,24,26)(18,21,28,20)(19,27,31,32)(23,29,30,25)(33,37)(34,41)(35,45)(36,42)(38,47)(39,48)(40,44)(43,46);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,23)(18,27)(19,20)(21,31)(22,29)(24,30)(25,26)(28,32)(33,38,39,35)(34,40,42,43)(36,46,41,44)(37,47,48,45)

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=g,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^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, 8044043);