#10481052

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

4171 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=i,f^2=g,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, 10388248);