#10455685

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

4063 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)(3,10,5,8)(6,9,14,12)(11,16,13,15)(20,28)(23,29)(26,32)(30,31)(34,39)(35,40)(37,45)(44,48);(1,3,7,5)(2,8,4,10)(6,11,14,13)(9,15,12,16)(17,18)(19,24)(20,29)(21,25)(22,27)(23,28)(26,30)(31,32)(33,34)(35,42)(36,40)(37,41)(38,39)(43,48)(44,47)(45,46);(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,32)(29,31)(33,35,38,40)(34,36,39,42)(37,43,45,47)(41,44,46,48);(1,4,7,2)(3,8,5,10)(6,12,14,9)(11,15,13,16)(17,20)(18,23)(19,26)(21,28)(22,30)(24,31)(25,29)(27,32)(33,34)(35,36)(37,41)(38,39)(40,42)(43,44)(45,46)(47,48);(1,5,7,3)(2,8,4,10)(6,11,14,13)(9,16,12,15)(17,19)(18,24)(20,26)(21,27)(22,25)(23,31)(28,32)(29,30)(33,36)(34,40)(35,39)(37,44)(38,42)(41,47)(43,46)(45,48);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,18)(19,22)(20,23)(21,25)(24,27)(26,30)(28,29)(31,32)(33,37,38,45)(34,41,39,46)(35,43,40,47)(36,44,42,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=gh,e^2=gh,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, 10362881);