#10455696

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	#52402	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4001 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:

(4,12)(7,13)(10,16)(14,15)(17,18,22,20)(19,25,27,23)(21,24,30,28)(26,32,29,31)(33,34,39,36)(35,42,38,40)(37,41,46,44)(43,48,45,47);(1,2)(3,8)(4,13)(5,9)(6,11)(7,12)(10,14)(15,16)(18,20)(23,25)(24,28)(31,32)(33,34,39,36)(35,40,38,42)(37,41,46,44)(43,47,45,48);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,16)(13,15)(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,32)(29,30)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(1,4)(2,7)(3,10)(5,12)(6,14)(8,15)(9,13)(11,16)(17,20,22,18)(19,25,27,23)(21,24,30,28)(26,31,29,32)(33,36,39,34)(35,42,38,40)(37,41,46,44)(43,47,45,48);(1,3)(2,8)(4,10)(5,11)(6,9)(7,15)(12,16)(13,14)(17,21,22,30)(18,24,20,28)(19,26,27,29)(23,31,25,32)(33,37,39,46)(34,41,36,44)(35,43,38,45)(40,47,42,48);(1,2)(3,6)(4,7)(5,9)(8,11)(10,14)(12,13)(15,16)(17,19)(18,23)(20,25)(21,29)(22,27)(24,32)(26,30)(28,31)(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=gh,c^2=gh,d^2=h,e^2=h,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, 10362892);