#8634721

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3884 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8541917);