#10349001

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

4126 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gi,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10256197);