#10446429

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

4123 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)(3,8)(4,10)(5,7)(6,9)(11,15)(12,16)(13,14)(17,18)(19,26)(20,30)(21,24)(22,31)(23,25)(27,32)(28,29)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,46)(47,48);(2,5)(8,10)(9,13)(15,16)(17,19,23,21)(18,24,25,26)(20,28,31,27)(22,29,30,32)(33,35)(34,36)(37,43)(38,39)(40,44)(41,42)(45,48)(46,47);(1,3,7,4)(2,8,5,10)(6,11,14,12)(9,15,13,16)(17,20)(18,22)(19,27)(21,28)(23,31)(24,32)(25,30)(26,29)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,47)(45,46);(1,4,7,3)(2,8,5,10)(6,11,14,12)(9,16,13,15)(17,20)(18,22)(19,28)(21,27)(23,31)(24,29)(25,30)(26,32)(33,34)(35,36)(37,45)(38,41)(39,42)(40,46)(43,48)(44,47);(1,5,7,2)(3,10,4,8)(6,13,14,9)(11,16,12,15)(17,21,23,19)(18,24,25,26)(20,28,31,27)(22,32,30,29)(33,36)(34,39)(35,41)(37,44)(38,42)(40,47)(43,45)(46,48);(1,6,7,14)(2,9,5,13)(3,11,4,12)(8,15,10,16)(17,22)(18,20)(19,29)(21,32)(23,30)(24,28)(25,31)(26,27)(33,37,38,46)(34,40,41,45)(35,43,39,47)(36,44,42,48)

Polycyclic presentation:

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

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, 10353625);