#8393317

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4082 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 8300513);