#10202032

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	#49083	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3965 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)(3,6)(4,9)(5,10)(7,8)(11,16)(12,14)(13,15)(18,22)(20,28)(23,25)(26,32)(33,34,39,43)(35,36,44,40)(37,45,47,41)(38,42,46,48);(1,3,7,11)(2,6,8,16)(4,12,13,5)(9,14,15,10)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35)(34,40)(36,43)(37,42)(38,45)(39,44)(41,46)(47,48);(2,8)(3,11)(4,13)(10,14)(17,18,21,22)(19,23,27,25)(20,24,28,29)(26,30,32,31)(33,36,39,40)(34,35,43,44)(37,46,47,38)(41,48,45,42);(1,4)(2,9)(3,12)(5,11)(6,14)(7,13)(8,15)(10,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,37,39,47)(34,41,43,45)(35,42,44,48)(36,38,40,46);(1,5,7,12)(2,10,8,14)(3,13,11,4)(6,15,16,9)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,38)(34,42)(35,41)(36,37)(39,46)(40,47)(43,48)(44,45);(1,6)(2,11)(3,8)(4,14)(5,9)(7,16)(10,13)(12,15)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,35)(34,40)(36,43)(37,42)(38,45)(39,44)(41,46)(47,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,d^2=hi,e^2=gh,f^2=ghi,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, 10109228);