#8366592

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	#58364	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3986 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8273788);