#8425449

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	#52370	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4146 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8332645);