#8137133

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	#48642	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3997 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=g,e^2=gh,f^2=gh,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, 8044329);