#8745945

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	#53750	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3943 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=hi,d^2=ghi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

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, 8653141);