#8843223

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3955 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)(3,4,10,8)(5,13,14,9)(11,15,16,12)(20,27)(23,25)(28,32)(30,31)(33,34,39,36)(35,41,37,40)(38,44,46,42)(43,47,45,48);(2,7)(4,8)(5,14)(11,16)(17,18,22,19)(20,23,27,25)(21,26,29,24)(28,31,32,30)(33,34,39,36)(35,41,37,40)(38,44,46,42)(43,47,45,48);(2,7)(3,10)(9,13)(11,16)(18,19)(23,25)(24,26)(30,31)(33,35,39,37)(34,40,36,41)(38,43,46,45)(42,47,44,48);(1,3,6,10)(2,8,7,4)(5,11,14,16)(9,15,13,12)(17,19,22,18)(20,23,27,25)(21,26,29,24)(28,30,32,31)(33,36,39,34)(35,40,37,41)(38,44,46,42)(43,47,45,48);(1,4,6,8)(2,3,7,10)(5,12,14,15)(9,11,13,16)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,37,39,35)(34,41,36,40)(38,45,46,43)(42,48,44,47);(1,5)(2,9)(3,11)(4,12)(6,14)(7,13)(8,15)(10,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,38,39,46)(34,42,36,44)(35,43,37,45)(40,47,41,48)

Polycyclic presentation:

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

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, 8750419);