#8828657

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	#57533	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3960 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8735853);