#10195490

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	#43253	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3930 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=ghi,d^2=gh,e^2=g,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, 10102686);