#8127328

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	#41495	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4022 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)(3,9)(4,12)(5,10)(6,7)(8,13)(11,14)(15,16)(21,29)(23,30)(26,32)(28,31)(34,39)(36,42)(40,48)(43,46);(2,7)(3,10)(8,12)(11,16)(17,18,22,24)(19,25,27,20)(21,23,29,30)(26,31,32,28)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(2,7)(3,10)(4,13)(14,15)(17,19,22,27)(18,20,24,25)(21,26,29,32)(23,28,30,31)(33,35,38,45)(34,40,39,48)(36,46,42,43)(37,47,41,44);(1,3)(2,5)(4,11)(6,10)(7,9)(8,14)(12,15)(13,16)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,31)(30,32)(33,36,38,42)(34,41,39,37)(35,43,45,46)(40,47,48,44);(1,4,6,13)(2,8,7,12)(3,11,10,16)(5,14,9,15)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,31)(30,32)(33,37,38,41)(34,42,39,36)(35,44,45,47)(40,46,48,43);(1,5)(2,3)(4,14)(6,9)(7,10)(8,11)(12,16)(13,15)(17,21)(18,23)(19,26)(20,28)(22,29)(24,30)(25,31)(27,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48)

Polycyclic presentation:

2^9|b^2=i,c^2=ghi,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8034524);