#8750420

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	#53071	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4013 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=i,e^2=g,f^2=gi,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, 8657616);