#10424746

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

4232 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,e^2=g,f^2=gh,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, 10331942);