#8395028

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	#50094	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4064 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)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(33,34,39,44)(35,41,36,40)(37,45,47,42)(38,43,48,46);(2,6)(3,10)(8,13)(11,16)(17,18,22,23)(19,25,28,24)(20,29,31,21)(26,32,30,27)(33,35,39,36)(34,40,44,41)(37,46,47,43)(38,45,48,42);(4,12)(8,13)(11,16)(14,15)(18,23)(21,29)(24,25)(27,32)(33,35,39,36)(34,40,44,41)(37,43,47,46)(38,42,48,45);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,19,22,28)(18,24,23,25)(20,30,31,26)(21,27,29,32)(33,36,39,35)(34,41,44,40)(37,43,47,46)(38,42,48,45);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,20)(18,21)(19,26)(22,31)(23,29)(24,32)(25,27)(28,30)(33,37,39,47)(34,42,44,45)(35,43,36,46)(38,40,48,41);(1,4,5,12)(2,8,6,13)(3,11,10,16)(7,14,9,15)(17,21)(18,20)(19,27)(22,29)(23,31)(24,30)(25,26)(28,32)(33,38)(34,43)(35,45)(36,42)(37,40)(39,48)(41,47)(44,46)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,d^2=hi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8302224);