#8458574

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4018 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)(19,23)(20,30)(21,26)(22,27)(24,28)(25,29)(31,32)(33,34,38,42)(35,39,37,41)(36,44,45,40)(43,48,46,47);(1,2)(3,7)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(17,19,24,22)(18,23,28,27)(20,31,29,21)(25,26,30,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,45)(46,48);(1,3)(2,5)(4,11)(6,9)(7,10)(8,13)(12,16)(14,15)(17,20)(18,25)(19,21)(22,31)(23,32)(24,29)(26,27)(28,30)(33,35,38,37)(34,39,42,41)(36,43,45,46)(40,47,44,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,21,24,31)(18,26,28,32)(19,29,22,20)(23,25,27,30)(33,36)(34,40)(35,43)(37,46)(38,45)(39,47)(41,48)(42,44);(1,4,6,12)(2,8,10,15)(3,11,9,16)(5,13,7,14)(17,22,24,19)(18,27,28,23)(20,31,29,21)(25,26,30,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,45)(46,48);(1,5,6,7)(2,9,10,3)(4,13,12,14)(8,16,15,11)(17,23,24,27)(18,19,28,22)(20,32,29,26)(21,30,31,25)(33,37,38,35)(34,41,42,39)(36,43,45,46)(40,47,44,48)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=h,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, 8365770);