#8372173

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4033 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)(3,5)(4,12)(6,7)(8,10)(9,13)(11,16)(14,15)(17,18,23,26)(19,21,20,24)(22,25,30,32)(27,29,28,31)(33,34,38,37)(35,39,43,41)(36,44,46,40)(42,48,45,47);(2,7)(3,10)(4,13)(15,16)(17,19,23,20)(18,24,26,21)(22,27,30,28)(25,31,32,29)(33,34,38,37)(35,41,43,39)(36,44,46,40)(42,47,45,48);(2,7)(5,8)(9,12)(15,16)(22,30)(25,32)(27,28)(29,31)(34,37)(39,41)(40,44)(47,48);(1,3)(2,8)(4,14)(5,7)(6,10)(9,15)(11,13)(12,16)(17,20,23,19)(18,21,26,24)(22,28,30,27)(25,29,32,31)(33,35,38,43)(34,39,37,41)(36,45,46,42)(40,48,44,47);(1,4,6,13)(2,9,7,12)(3,11,10,14)(5,15,8,16)(17,21)(18,20)(19,26)(22,29)(23,24)(25,28)(27,32)(30,31)(33,36,38,46)(34,40,37,44)(35,42,43,45)(39,47,41,48);(1,5,6,8)(2,3,7,10)(4,15,13,16)(9,11,12,14)(17,22,23,30)(18,25,26,32)(19,27,20,28)(21,29,24,31)(33,37,38,34)(35,41,43,39)(36,44,46,40)(42,48,45,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=gh,e^2=h,f^2=gi,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, 8279369);