#8186904

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	#43155	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4089 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)(3,10)(4,7)(5,13)(6,9)(8,14)(11,15)(12,16)(18,23)(24,25)(26,30)(31,32)(33,34,38,42)(35,43,45,39)(36,46,44,40)(37,41,47,48);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,15)(13,14)(17,18,22,23)(19,24,20,25)(21,26,29,30)(27,31,28,32)(33,35,38,45)(34,39,42,43)(36,37,44,47)(40,41,46,48);(1,3,6,4)(2,7,9,10)(5,11,14,12)(8,15,13,16)(17,19,22,20)(18,24,23,25)(21,28,29,27)(26,32,30,31)(33,36)(34,40)(35,37)(38,44)(39,41)(42,46)(43,48)(45,47);(1,4,6,3)(2,7,9,10)(5,12,14,11)(8,15,13,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(33,34)(35,39)(36,40)(37,41)(38,42)(43,45)(44,46)(47,48);(1,5,6,14)(2,8,9,13)(3,11,4,12)(7,15,10,16)(17,20,22,19)(18,25,23,24)(21,28,29,27)(26,32,30,31)(33,34)(35,39)(36,40)(37,41)(38,42)(43,45)(44,46)(47,48);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,15)(13,14)(17,21)(18,26)(19,27)(20,28)(22,29)(23,30)(24,31)(25,32)(33,37)(34,41)(35,44)(36,45)(38,47)(39,46)(40,43)(42,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,d^2=i,e^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, 8094100);