#8188039

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	#46478	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4067 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)(4,8)(5,6)(7,9)(11,16)(12,15)(13,14)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29)(33,34)(35,41)(36,37)(38,42)(39,40)(43,48)(44,45)(46,47);(1,3,7,4)(2,8,9,10)(5,12,16,14)(6,13,11,15)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31)(34,40)(36,41)(42,47)(44,48);(2,9)(6,11)(8,10)(13,15)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31)(33,35,39,37)(34,41,40,36)(38,45,46,43)(42,44,47,48);(1,4,7,3)(2,10,9,8)(5,14,16,12)(6,15,11,13)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31)(33,36)(34,35)(37,40)(38,44)(39,41)(42,43)(45,47)(46,48);(1,5)(2,11)(3,12)(4,14)(6,9)(7,16)(8,15)(10,13)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(33,37,39,35)(34,36,40,41)(38,45,46,43)(42,44,47,48);(1,6,7,11)(2,5,9,16)(3,13,4,15)(8,12,10,14)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)(33,38)(34,42)(35,43)(36,44)(37,45)(39,46)(40,47)(41,48)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,d^2=hi,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, 8095235);