#8505461

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	#50835	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4044 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:

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

Polycyclic presentation:

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

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, 8412657);