#8841372

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3882 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gh,d^2=hi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

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, 8748568);