#10166174

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	#51166	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3897 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,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, 10073370);