#8750736

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,d^2=i,e^2=g,f^2=gi,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, 8657932);