#8806850

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	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3971 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)(3,7)(4,12)(5,6)(8,13)(9,10)(11,15)(14,16)(17,18,23,20)(19,26,22,24)(21,25,29,28)(27,32,30,31)(33,34,39,37)(35,42,41,38)(36,44,46,40)(43,45,47,48);(1,2)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(21,29)(25,28)(27,30)(31,32)(34,37)(35,41)(40,44)(43,47);(2,6)(3,10)(8,12)(11,16)(17,19,23,22)(18,24,20,26)(21,27,29,30)(25,31,28,32)(33,35)(34,38)(36,43)(37,42)(39,41)(40,45)(44,48)(46,47);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20,23,18)(19,26,22,24)(21,28,29,25)(27,32,30,31)(33,36)(34,40)(35,43)(37,44)(38,45)(39,46)(41,47)(42,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,21,23,29)(18,25,20,28)(19,27,22,30)(24,31,26,32)(33,37,39,34)(35,42,41,38)(36,44,46,40)(43,48,47,45);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,22,23,19)(18,26,20,24)(21,30,29,27)(25,32,28,31)(33,38,39,42)(34,41,37,35)(36,45,46,48)(40,47,44,43)

Polycyclic presentation:

2^9|a^2=ghi,c^2=hi,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, 8714046);