#8905142

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3920 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8812338);