#8957592

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	#52048	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3991 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=h,d^2=ghi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=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, 8864788);