#10412069

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4079 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=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, 10319265);