#10168437

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	#50791	C₂
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:

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

Polycyclic presentation:

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

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, 10075633);