#8180488

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3982 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=g,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8087684);