#8458429

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	#42533	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4085 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:

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

Polycyclic presentation:

2^9|c^2=gh,e^2=i,f^2=gi,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, 8365625);