#8901410

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3923 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:

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

Polycyclic presentation:

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

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, 8808606);