#8901221

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

3912 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=ghi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,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, 8808417);