#8436672

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

4013 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gi,d^2=g,e^2=h,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, 8343868);