#10216651

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3912 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)(19,26)(20,24)(21,29)(22,23)(25,31)(27,28)(30,32)(33,34,40,44)(35,39,37,42)(36,46,45,41)(38,43,47,48);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(18,23)(20,26)(25,29)(28,30)(33,35,40,37)(34,39,44,42)(36,47,45,38)(41,43,46,48);(17,19,22,24)(18,20,23,26)(21,27,31,32)(25,28,29,30)(33,36,40,45)(34,41,44,46)(35,38,37,47)(39,48,42,43);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,19)(18,20)(21,27)(22,24)(23,26)(25,28)(29,30)(31,32)(33,37,40,35)(34,42,44,39)(36,38,45,47)(41,48,46,43);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,20)(18,24)(19,23)(21,30)(22,26)(25,27)(28,31)(29,32)(33,38)(34,43)(35,45)(36,37)(39,41)(40,47)(42,46)(44,48);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,21,22,31)(18,25,23,29)(19,27,24,32)(20,28,26,30)(33,39,40,42)(34,35,44,37)(36,48,45,43)(38,46,47,41)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,d^2=i,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, 10123847);