#8434124

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	#52372	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4263 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)(3,10,6,9)(4,7,5,15)(12,16,13,14)(20,28)(24,31)(27,32)(29,30)(34,40)(35,44)(36,46)(43,48);(1,3,8,6)(2,9,11,10)(4,12,5,13)(7,14,15,16)(17,18,22,26)(19,21,25,23)(20,24,28,31)(27,29,32,30)(33,34)(35,41)(36,47)(37,43)(38,44)(39,40)(42,46)(45,48);(1,4,8,5)(2,7,11,15)(3,12,6,13)(9,14,10,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,35)(34,41)(36,45)(37,46)(38,40)(39,44)(42,43)(47,48);(1,5,8,4)(2,7,11,15)(3,13,6,12)(9,14,10,16)(17,19)(18,21)(20,27)(22,25)(23,26)(24,29)(28,32)(30,31)(33,36,39,46)(34,42,40,47)(35,45,44,37)(38,48,41,43);(1,6,8,3)(2,10,11,9)(4,13,5,12)(7,16,15,14)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,32)(26,31)(33,37,39,45)(34,43,40,48)(35,46,44,36)(38,47,41,42);(1,7)(2,5)(3,14)(4,11)(6,16)(8,15)(9,13)(10,12)(17,21)(18,25)(19,26)(20,29)(22,23)(24,32)(27,31)(28,30)(33,38)(34,44)(35,40)(36,43)(37,47)(39,41)(42,45)(46,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=g,e^2=g,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, 8341320);