#8899625

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	#46398	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4116 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)(3,9)(4,8)(5,10)(6,7)(11,16)(12,15)(13,14)(17,18,22,21)(19,23,25,27)(20,24,28,29)(26,30,31,32)(33,34,38,41)(35,42,37,39)(36,40,44,48)(43,47,45,46);(1,3,7,4)(2,8,6,9)(5,12,14,11)(10,16,13,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(35,37)(39,42)(43,45)(46,47);(2,6)(8,9)(10,13)(15,16)(19,25)(23,27)(26,31)(30,32)(33,35,38,37)(34,39,41,42)(36,43,44,45)(40,46,48,47);(1,4,7,3)(2,9,6,8)(5,11,14,12)(10,15,13,16)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32)(33,36)(34,40)(35,43)(37,45)(38,44)(39,46)(41,48)(42,47);(1,5,7,14)(2,10,6,13)(3,11,4,12)(8,15,9,16)(17,21,22,18)(19,27,25,23)(20,24,28,29)(26,30,31,32)(33,37,38,35)(34,39,41,42)(36,43,44,45)(40,47,48,46);(1,6,7,2)(3,9,4,8)(5,13,14,10)(11,16,12,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,34)(35,42)(36,40)(37,39)(38,41)(43,47)(44,48)(45,46)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=g,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, 8806821);