#10166538

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	#55986	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3984 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)(35,43)(36,38)(37,40)(39,42)(41,48)(44,45)(46,47);(1,3)(2,8)(4,12)(5,11)(6,13)(7,14)(9,16)(10,15)(19,25)(22,29)(26,31)(30,32)(35,37)(40,43)(44,46)(45,47);(17,18,21,24)(19,22,25,29)(20,23,28,27)(26,30,31,32)(33,35,39,37)(34,40,42,43)(36,46,41,44)(38,45,48,47);(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)(33,36,39,41)(34,38,42,48)(35,44,37,46)(40,47,43,45);(1,5,7,2)(3,11,14,8)(4,9,13,15)(6,10,12,16)(17,19)(18,22)(20,26)(21,25)(23,30)(24,29)(27,32)(28,31)(33,37,39,35)(34,40,42,43)(36,44,41,46)(38,45,48,47);(1,6,7,12)(2,10,5,16)(3,13,14,4)(8,15,11,9)(17,20)(18,23)(19,26)(21,28)(22,30)(24,27)(25,31)(29,32)(33,38)(34,41)(35,45)(36,42)(37,47)(39,48)(40,44)(43,46)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=gi,e^2=gi,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, 10073734);