#8460641

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3963 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8367837);