#10455669

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3845 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=gh,e^2=h,f^2=h,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, 10362865);