#10166055

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	#2431	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3928 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gi,d^2=h,e^2=ghi,f^2=gh,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, 10073251);