#10534312

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	#55105	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4136 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)(5,8)(6,9)(7,10)(11,12)(13,15)(14,16)(18,23)(19,25)(21,29)(28,31)(33,34,40,38)(35,41,46,45)(36,47,37,42)(39,48,44,43);(1,3,7,5)(2,8,10,4)(6,11,14,13)(9,15,16,12)(17,18,22,23)(19,20,25,24)(21,26,29,30)(27,28,32,31)(33,35,40,46)(34,41,38,45)(36,44,37,39)(42,48,47,43);(17,19)(18,24)(20,23)(21,27)(22,25)(26,31)(28,30)(29,32)(33,36,40,37)(34,42,38,47)(35,44,46,39)(41,48,45,43);(1,4,7,8)(2,3,10,5)(6,12,14,15)(9,11,16,13)(17,19)(18,20)(21,27)(22,25)(23,24)(26,28)(29,32)(30,31)(33,37,40,36)(34,42,38,47)(35,39,46,44)(41,48,45,43);(1,5,7,3)(2,8,10,4)(6,11,14,13)(9,12,16,15)(17,20)(18,25)(19,23)(21,28)(22,24)(26,32)(27,30)(29,31)(33,38,40,34)(35,45,46,41)(36,47,37,42)(39,48,44,43);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,21,22,29)(18,26,23,30)(19,27,25,32)(20,28,24,31)(33,39)(34,43)(35,36)(37,46)(38,48)(40,44)(41,42)(45,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=hi,d^2=gh,e^2=gh,f^2=i,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, 10441508);