#8806885

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

3833 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=ghi,d^2=hi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8714081);