#8174362

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3908 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=hi,d^2=h,e^2=h,f^2=i,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, 8081558);