#8772641

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	#43189	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3939 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(17,18,23,24)(19,27,28,25)(20,30,29,26)(21,22,31,32)(34,39)(37,40)(41,46)(45,47);(2,6)(4,13)(7,9)(11,16)(18,24)(19,28)(21,31)(26,30)(36,44)(41,46)(43,48)(45,47);(2,6)(3,10)(8,12)(11,16)(17,19,23,28)(18,25,24,27)(20,21,29,31)(22,30,32,26)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,20,23,29)(18,26,24,30)(19,21,28,31)(22,27,32,25)(33,35,38,42)(34,40,39,37)(36,43,44,48)(41,47,46,45);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,21)(18,22)(19,29)(20,28)(23,31)(24,32)(25,30)(26,27)(33,36,38,44)(34,41,39,46)(35,43,42,48)(37,45,40,47);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,22)(18,21)(19,30)(20,27)(23,32)(24,31)(25,29)(26,28)(33,37,38,40)(34,42,39,35)(36,45,44,47)(41,48,46,43)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8679837);