#10038473

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3882 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=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, 9945669);