#10324628

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	#53165	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3988 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)(3,4)(5,13)(6,7)(8,15)(9,12)(10,16)(11,14)(17,18,22,26)(19,23,27,32)(20,24,30,29)(21,25,31,28)(34,40)(35,36)(38,48)(43,46);(1,3,6,9)(2,4,7,12)(5,14,15,10)(8,16,13,11)(19,27)(21,31)(23,32)(25,28)(33,34)(35,41)(36,45)(37,38)(39,40)(42,48)(43,47)(44,46);(2,7)(3,9)(8,13)(10,14)(17,19)(18,23)(20,25)(21,29)(22,27)(24,31)(26,32)(28,30)(33,35,39,36)(34,41,40,45)(37,46,42,43)(38,44,48,47);(1,4)(2,3)(5,11)(6,12)(7,9)(8,10)(13,14)(15,16)(17,19)(18,23)(20,25)(21,29)(22,27)(24,31)(26,32)(28,30)(33,36,39,35)(34,41,40,45)(37,46,42,43)(38,47,48,44);(1,2)(3,4)(5,8)(6,7)(9,12)(10,11)(13,15)(14,16)(17,20)(18,24)(19,28)(21,23)(22,30)(25,27)(26,29)(31,32)(33,37,39,42)(34,38,40,48)(35,43,36,46)(41,47,45,44);(1,5)(2,8)(3,10)(4,11)(6,15)(7,13)(9,14)(12,16)(17,21)(18,25)(19,29)(20,23)(22,31)(24,27)(26,28)(30,32)(33,38)(34,42)(35,44)(36,47)(37,40)(39,48)(41,43)(45,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10231824);