#8374189

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	#42079	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4004 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=gh,e^2=gh,f^2=ghi,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, 8281385);