#8581497

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×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3915 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=ghi,d^2=hi,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8488693);