#8772447

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4015 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=ghi,e^2=gh,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, 8679643);