#8745883

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	#42094	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3949 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=i,d^2=hi,e^2=g,f^2=gh,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, 8653079);