#8739362

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

3735 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)(3,11)(4,10)(5,8)(6,12)(7,9)(13,16)(14,15)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,34)(35,40)(36,42)(37,41)(38,43)(39,44)(45,46)(47,48);(1,3,7,5)(2,8,9,11)(4,15,12,13)(6,14,10,16)(18,22)(23,25)(24,29)(30,31)(33,35,39,37)(34,40,44,41)(36,47,43,45)(38,46,42,48);(2,9)(4,12)(8,11)(13,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,35,39,37)(34,41,44,40)(36,45,43,47)(38,46,42,48);(1,4)(2,10)(3,13)(5,15)(6,9)(7,12)(8,14)(11,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,36,39,43)(34,42,44,38)(35,45,37,47)(40,46,41,48);(1,5,7,3)(2,11,9,8)(4,15,12,13)(6,14,10,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,37,39,35)(34,40,44,41)(36,47,43,45)(38,48,42,46);(1,6,7,10)(2,12,9,4)(3,14,5,16)(8,13,11,15)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,38,39,42)(34,43,44,36)(35,46,37,48)(40,45,41,47)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gh,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8646558);