#10412110

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4205 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^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, 10319306);