#8180471

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	#40426	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4063 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)(3,9)(4,5)(6,10)(7,8)(11,16)(12,13)(14,15)(18,23)(19,26)(20,28)(29,31)(33,34,39,38)(35,40,45,44)(36,46,41,37)(42,48,47,43);(2,8)(4,9)(10,15)(12,16)(17,18,22,23)(19,21,26,24)(20,25,28,30)(27,29,32,31)(33,35,39,45)(34,40,38,44)(36,42,41,47)(37,43,46,48);(1,3,7,5)(2,9,8,4)(6,13,14,11)(10,12,15,16)(17,19,22,26)(18,24,23,21)(20,27,28,32)(25,31,30,29)(33,36)(34,37)(35,42)(38,46)(39,41)(40,43)(44,48)(45,47);(1,4)(2,3)(5,8)(6,12)(7,9)(10,11)(13,15)(14,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,30)(29,32)(33,35)(34,40)(36,42)(37,43)(38,44)(39,45)(41,47)(46,48);(1,5,7,3)(2,4,8,9)(6,13,14,11)(10,12,15,16)(17,20)(18,25)(19,27)(21,29)(22,28)(23,30)(24,31)(26,32)(33,37,39,46)(34,41,38,36)(35,43,45,48)(40,47,44,42);(1,6)(2,10)(3,11)(4,12)(5,13)(7,14)(8,15)(9,16)(17,21,22,24)(18,26,23,19)(20,29,28,31)(25,32,30,27)(33,38,39,34)(35,44,45,40)(36,46,41,37)(42,48,47,43)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,e^2=gh,f^2=h,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, 8087667);