#8371553

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	#39906	C₂
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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8278749);