#10412234

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	#56607	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4109 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10319430);