#8541907

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

3809 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,d^2=gh,e^2=ghi,f^2=gh,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, 8449103);