#10181206

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3945 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=gi,d^2=i,e^2=ghi,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, 10088402);