#10480819

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4011 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10388015);