#10217005

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	#53526	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3980 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)(4,7,11,9)(5,10,13,8)(12,15,16,14)(17,18,22,23)(19,25,26,20)(21,29,31,24)(27,28,32,30)(33,34)(35,40)(36,42)(37,43)(38,47)(39,41)(44,45)(46,48);(4,11)(7,9)(12,16)(14,15)(18,23)(20,25)(24,29)(28,30)(33,35,39,43)(34,40,41,37)(36,45,46,38)(42,44,48,47);(2,3)(7,9)(8,10)(14,15)(18,23)(19,26)(24,29)(27,32)(34,41)(35,43)(36,46)(44,47);(1,3,6,2)(4,9,11,7)(5,10,13,8)(12,15,16,14)(17,19,22,26)(18,20,23,25)(21,27,31,32)(24,28,29,30)(33,36)(34,42)(35,45)(37,47)(38,43)(39,46)(40,44)(41,48);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,20)(18,19)(21,30)(22,25)(23,26)(24,32)(27,29)(28,31)(33,37)(34,43)(35,41)(36,44)(38,48)(39,40)(42,45)(46,47);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,21,22,31)(18,24,23,29)(19,27,26,32)(20,28,25,30)(33,38,39,45)(34,44,41,47)(35,36,43,46)(37,42,40,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,d^2=hi,e^2=ghi,f^2=gi,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, 10124201);