#8899631

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

3986 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,d^2=g,e^2=gi,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, 8806827);