#8188339

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3897 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)(17,18,23,25)(19,26,20,22)(21,29,31,24)(27,30,28,32)(33,34,38,42)(35,40,36,39)(37,41,46,45)(43,48,44,47);(1,3,6,11)(2,7,5,9)(4,10,13,16)(8,14,12,15)(17,18,23,25)(19,26,20,22)(21,24,31,29)(27,32,28,30)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,48)(45,46);(1,3)(2,7)(4,10)(5,9)(6,11)(8,14)(12,15)(13,16)(17,19,23,20)(18,22,25,26)(21,27,31,28)(24,30,29,32)(33,34)(35,39)(36,40)(37,45)(38,42)(41,46)(43,48)(44,47);(1,3)(2,7)(4,10)(5,9)(6,11)(8,14)(12,15)(13,16)(17,20,23,19)(18,22,25,26)(21,28,31,27)(24,30,29,32)(33,35,38,36)(34,39,42,40)(37,43,46,44)(41,47,45,48);(1,4,6,13)(2,8,5,12)(3,10,11,16)(7,14,9,15)(17,21,23,31)(18,24,25,29)(19,27,20,28)(22,30,26,32)(33,36,38,35)(34,40,42,39)(37,44,46,43)(41,48,45,47);(1,5,6,2)(3,9,11,7)(4,12,13,8)(10,15,16,14)(17,22)(18,19)(20,25)(21,30)(23,26)(24,27)(28,29)(31,32)(33,37,38,46)(34,41,42,45)(35,43,36,44)(39,47,40,48)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,d^2=gi,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, 8095535);