#10216639

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

3780 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)(19,27)(20,30)(21,26)(22,24)(23,25)(28,32)(29,31)(33,34,39,40)(35,37,42,47)(36,41,45,48)(38,46,44,43);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,19,23,22)(18,24,25,27)(20,28,26,31)(21,32,30,29)(33,35)(34,37)(36,44)(38,45)(39,42)(40,47)(41,43)(46,48);(18,25)(20,26)(24,27)(28,31)(34,40)(35,42)(36,45)(43,46);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,20)(18,21)(19,28)(22,31)(23,26)(24,32)(25,30)(27,29)(33,36,39,45)(34,41,40,48)(35,44,42,38)(37,43,47,46);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,21)(18,26)(19,29)(20,25)(22,32)(23,30)(24,28)(27,31)(33,37)(34,42)(35,40)(36,46)(38,48)(39,47)(41,44)(43,45);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,22,23,19)(18,27,25,24)(20,31,26,28)(21,29,30,32)(33,38,39,44)(34,43,40,46)(35,45,42,36)(37,41,47,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,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, 10123835);