#8528877

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	#2825	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3884 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)(3,8)(4,11)(5,9)(6,7)(10,16)(12,13)(14,15)(17,18,22,25)(19,23,28,31)(20,21,24,29)(26,27,30,32)(33,34)(35,43)(36,47)(37,48)(38,42)(39,40)(41,44)(45,46);(1,2)(3,4)(5,9)(6,7)(8,11)(10,12)(13,16)(14,15)(17,19)(18,23)(20,26)(21,27)(22,28)(24,30)(25,31)(29,32)(34,40)(35,44)(36,48)(42,46);(2,7)(4,8)(9,15)(12,16)(17,19,22,28)(18,23,25,31)(20,26,24,30)(21,27,29,32)(33,35,39,44)(34,41,40,43)(36,45,48,38)(37,46,47,42);(1,3)(2,8)(4,7)(5,13)(6,11)(9,12)(10,14)(15,16)(17,20)(18,21)(19,26)(22,24)(23,27)(25,29)(28,30)(31,32)(33,36)(34,37)(35,45)(38,44)(39,48)(40,47)(41,46)(42,43);(1,4,6,8)(2,3,7,11)(5,12,14,16)(9,10,15,13)(17,19)(18,23)(20,26)(21,27)(22,28)(24,30)(25,31)(29,32)(33,37)(34,36)(35,46)(38,43)(39,47)(40,48)(41,45)(42,44);(1,5)(2,9)(3,10)(4,12)(6,14)(7,15)(8,16)(11,13)(17,21)(18,24)(19,27)(20,25)(22,29)(23,30)(26,31)(28,32)(33,38,39,45)(34,42,40,46)(35,36,44,48)(37,43,47,41)

Polycyclic presentation:

2^9|a^2=hi,c^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8436073);