#8818196

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	#50088	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4000 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)(3,6)(4,12)(5,8)(7,13)(9,11)(10,15)(14,16)(33,34,40,46)(35,43,37,41)(36,45,39,42)(38,44,48,47);(1,2)(3,9)(4,12)(5,8)(6,11)(7,13)(10,14)(15,16)(17,18)(19,22)(20,28)(21,24)(23,29)(25,27)(26,31)(30,32)(33,35,40,37)(34,41,46,43)(36,47,39,44)(38,45,48,42);(1,2,5,8)(3,9,11,6)(4,7,13,12)(10,15,16,14)(17,18,21,24)(19,25,27,22)(20,23,29,28)(26,31,32,30)(33,36,40,39)(34,42,46,45)(35,47,37,44)(38,43,48,41);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,13)(15,16)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,30,28,31)(33,37,40,35)(34,43,46,41)(36,47,39,44)(38,45,48,42);(1,3,5,11)(2,6,8,9)(4,10,13,16)(7,14,12,15)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,29)(31,32)(33,38,40,48)(34,44,46,47)(35,42,37,45)(36,41,39,43);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)(33,39,40,36)(34,45,46,42)(35,47,37,44)(38,43,48,41)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,d^2=hi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,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, 8725392);