#8107675

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	#40852	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4045 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=i,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014871);