#484041

order	512
nilpotency class	4
isoclinism	isoclinic class of #484039
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	#2186	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1451 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

416 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216779, #72216788, #72216926, …
maximal normal subgroup of	#72216779, #72216788, #72216926, …
maximal quotient of	C₂×#484041, #72318536, #72318584, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391237);