#10455710

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

4003 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:

(1,2,7,8)(3,9,5,4)(6,10,14,15)(11,16,13,12)(17,18)(19,24)(20,22)(21,28)(23,27)(25,29)(26,30)(31,32)(36,44)(39,45)(42,48)(46,47);(2,8)(4,9)(10,15)(12,16)(17,18,22,20)(19,24,27,23)(21,25,29,28)(26,31,32,30)(33,34)(35,40)(36,45)(37,41)(38,43)(39,44)(42,46)(47,48);(1,3,7,5)(2,9,8,4)(6,11,14,13)(10,16,15,12)(17,19)(18,23)(20,24)(21,26)(22,27)(25,30)(28,31)(29,32)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,48)(45,47);(1,4)(2,3)(5,8)(6,12)(7,9)(10,11)(13,15)(14,16)(17,20,22,18)(19,24,27,23)(21,28,29,25)(26,31,32,30)(33,36)(34,39)(35,42)(37,44)(38,46)(40,47)(41,45)(43,48);(1,5,7,3)(2,4,8,9)(6,11,14,13)(10,16,15,12)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32)(33,35)(34,40)(36,42)(37,43)(38,41)(39,47)(44,48)(45,46);(1,6,7,14)(2,10,8,15)(3,11,5,13)(4,12,9,16)(17,21)(18,25)(19,26)(20,28)(22,29)(23,30)(24,31)(27,32)(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=h,c^2=gh,d^2=h,e^2=gh,f^2=gh,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, 10362906);