#10216699

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3777 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,d^2=hi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10123895);