#10504487

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	#57549	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3898 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)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)(17,18,24,27)(19,28,22,25)(20,23,21,26)(29,30,31,32)(33,34,39,36)(35,42,38,40)(37,41,46,44)(43,48,45,47);(2,6)(7,13)(8,14)(15,16)(17,19,24,22)(18,25,27,28)(20,31,21,29)(23,30,26,32)(33,34,39,36)(35,40,38,42)(37,41,46,44)(43,47,45,48);(17,20,24,21)(18,26,27,23)(19,29,22,31)(25,30,28,32)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(1,2)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)(17,21,24,20)(18,23,27,26)(19,29,22,31)(25,30,28,32)(33,36,39,34)(35,42,38,40)(37,41,46,44)(43,47,45,48);(1,3,5,10)(2,7,6,13)(4,11,12,9)(8,16,14,15)(17,22,24,19)(18,25,27,28)(20,31,21,29)(23,32,26,30)(33,37,39,46)(34,41,36,44)(35,43,38,45)(40,47,42,48);(1,4,5,12)(2,8,6,14)(3,9,10,11)(7,15,13,16)(17,23)(18,20)(19,30)(21,27)(22,32)(24,26)(25,31)(28,29)(33,38,39,35)(34,42,36,40)(37,45,46,43)(41,48,44,47)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=h,e^2=ghi,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, 10411683);