#8186910

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	#42894	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4077 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:

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

Polycyclic presentation:

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

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, 8094106);