#8188331

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×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3983 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)(33,34,40,39)(35,43,38,41)(36,46,42,37)(44,45,47,48);(1,3)(2,7)(4,10)(5,9)(6,11)(8,14)(12,15)(13,16)(17,18,23,25)(19,26,20,22)(21,24,31,29)(27,32,28,30)(33,35,40,38)(34,41,39,43)(36,47,42,44)(37,48,46,45);(2,5)(7,9)(8,12)(14,15)(17,19,23,20)(18,22,25,26)(21,27,31,28)(24,30,29,32)(33,36)(34,37)(35,44)(38,47)(39,46)(40,42)(41,45)(43,48);(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)(33,37,40,46)(34,42,39,36)(35,45,38,48)(41,47,43,44);(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)(33,38,40,35)(34,43,39,41)(36,47,42,44)(37,48,46,45);(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)(33,39,40,34)(35,43,38,41)(36,46,42,37)(44,48,47,45)

Polycyclic presentation:

2^9|a^2=h,b^2=i,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, 8095527);