#10455707

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	#52435	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3902 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)(3,6)(4,7)(5,8)(9,13)(10,14)(11,15)(12,16)(17,18,23,27)(19,26,30,24)(20,22,21,25)(28,29,31,32)(36,44)(39,45)(42,48)(46,47);(1,2,5,8)(3,6,10,14)(4,7,12,16)(9,13,11,15)(17,19)(18,24)(20,31)(21,28)(22,29)(23,30)(25,32)(26,27)(33,34)(35,40)(36,45)(37,41)(38,43)(39,44)(42,46)(47,48);(17,20,23,21)(18,25,27,22)(19,28,30,31)(24,29,26,32)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,48)(45,47);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,15)(12,16)(17,21,23,20)(18,22,27,25)(19,28,30,31)(24,29,26,32)(33,36)(34,39)(35,42)(37,44)(38,46)(40,47)(41,45)(43,48);(1,3,5,10)(2,6,8,14)(4,11,12,9)(7,15,16,13)(17,19)(18,26)(20,28)(21,31)(22,29)(23,30)(24,27)(25,32)(33,35)(34,40)(36,42)(37,43)(38,41)(39,47)(44,48)(45,46);(1,4,5,12)(2,7,8,16)(3,9,10,11)(6,13,14,15)(17,22)(18,20)(19,29)(21,27)(23,25)(24,31)(26,28)(30,32)(33,34)(35,38)(36,39)(37,41)(40,43)(42,46)(44,45)(47,48)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,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, 10362903);