#8159022

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	#2539	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3984 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=g,e^2=h,f^2=gh,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^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, 8066218);