#10106953

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	#44749	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3914 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=hi,d^2=gh,e^2=g,f^2=g,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, 10014149);