#483931

order	512
nilpotency class	4
isoclinism	isoclinic class of #483928
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#33675	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

417 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fgi,d^2=i,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cg,d^a=dh,e^a=eg,g^a=gi,c^b=ch,e^b=egi,f^b=fi,g^b=gi,d^c=di,e^c=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286391, #72286482, #72286537, …
maximal normal subgroup of	#72286391, #72286482, #72286537, …
maximal quotient of	#72343402, #72343440, #72343468, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391127);