#10195917

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	#48242	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3916 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)(3,10)(4,13)(5,12)(6,8)(7,14)(9,15)(11,16)(17,18)(19,24)(20,22)(21,26)(23,25)(27,32)(28,31)(29,30)(33,34,40,38)(35,43,39,41)(36,42,37,46)(44,48,45,47);(1,3,6,14)(2,7,8,10)(4,12,15,11)(5,13,16,9)(17,19)(18,24)(20,29)(21,27)(22,30)(23,28)(25,31)(26,32)(33,35,40,39)(34,41,38,43)(36,44,37,45)(42,47,46,48);(2,8)(4,15)(7,10)(11,12)(18,25)(19,28)(20,27)(26,30)(33,36,40,37)(34,42,38,46)(35,44,39,45)(41,47,43,48);(1,4)(2,9)(3,12)(5,7)(6,15)(8,13)(10,16)(11,14)(17,20,23,27)(18,22,25,32)(19,29,28,21)(24,30,31,26)(33,37,40,36)(34,42,38,46)(35,45,39,44)(41,47,43,48);(1,3)(2,10)(4,11)(5,9)(6,14)(7,8)(12,15)(13,16)(17,21,23,29)(18,26,25,30)(19,20,28,27)(22,31,32,24)(33,38,40,34)(35,43,39,41)(36,46,37,42)(44,48,45,47);(1,5)(2,11)(3,13)(4,7)(6,16)(8,12)(9,14)(10,15)(17,22)(18,27)(19,30)(20,25)(21,24)(23,32)(26,28)(29,31)(33,39,40,35)(34,41,38,43)(36,45,37,44)(42,47,46,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=ghi,d^2=gh,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, 10103113);