#8186980

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	#39004	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4099 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)(3,4)(5,8)(6,14)(7,10)(9,15)(11,13)(12,16)(17,18)(19,21)(20,28)(22,23)(24,26)(25,30)(27,31)(29,32)(33,34,40,39)(35,45,47,41)(36,48,46,42)(37,44,38,43);(1,3,7,8)(2,4,10,5)(6,11,15,16)(9,12,14,13)(18,23)(19,26)(25,28)(27,32)(33,35)(34,41)(36,37)(38,46)(39,45)(40,47)(42,44)(43,48);(1,4,7,5)(2,8,10,3)(6,12,15,13)(9,16,14,11)(17,18,22,23)(19,24,26,21)(20,25,30,28)(27,31,32,29)(33,36,40,46)(34,42,39,48)(35,37,47,38)(41,44,45,43);(1,5,7,4)(2,8,10,3)(6,13,15,12)(9,16,14,11)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,29)(30,32)(33,37,40,38)(34,43,39,44)(35,36,47,46)(41,48,45,42);(1,6,7,15)(2,9,10,14)(3,11,8,16)(4,12,5,13)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31)(33,38,40,37)(34,44,39,43)(35,46,47,36)(41,42,45,48);(1,2)(3,4)(5,8)(6,9)(7,10)(11,12)(13,16)(14,15)(17,21,22,24)(18,19,23,26)(20,29,30,31)(25,27,28,32)(33,39,40,34)(35,45,47,41)(36,42,46,48)(37,44,38,43)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,d^2=gi,e^2=h,f^2=g,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, 8094176);