#8551026

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

3749 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)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,34,38,41)(35,36,43,39)(37,45,46,40)(42,47,48,44);(1,3,7,10)(2,8,6,5)(4,11,13,16)(9,15,14,12)(17,18,21,24)(19,22,27,25)(20,23,28,30)(26,29,32,31)(33,35,38,43)(34,39,41,36)(37,42,46,48)(40,47,45,44);(3,10)(5,8)(11,16)(12,15)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,45,48);(1,4)(2,9)(3,11)(5,12)(6,14)(7,13)(8,15)(10,16)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,45)(46,48);(1,5)(2,3)(4,12)(6,10)(7,8)(9,11)(13,15)(14,16)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,45)(43,48);(1,6,7,2)(3,5,10,8)(4,9,13,14)(11,15,16,12)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=h,d^2=gh,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^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, 8458222);