#484049

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	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1451 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

407 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)(17,18,22,27,23,28,20,25)(19,29,30,46,33,47,32,43)(21,35,38,44,37,42,34,26)(24,39,40,31,41,36,45,48);(17,19,23,33)(18,24,28,41)(20,30,22,32)(21,36,37,39)(25,40,27,45)(26,43,44,46)(29,35,47,42)(31,34,48,38);(1,3,4,10,5,9,11,15)(2,6,7,13,8,12,14,16)(18,25,28,27)(21,37)(24,40,41,45)(26,35,44,42)(29,43,47,46)(31,48)(34,38)(36,39);(3,9)(6,12)(10,15)(13,16)(17,20,23,22)(18,25,28,27)(19,30,33,32)(21,34,37,38)(24,40,41,45)(26,42,44,35)(29,46,47,43)(31,36,48,39)

Polycyclic presentation:

2^9|a^2=f,b^2=i,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=dhi,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287372, #72321460, #72321548, …
maximal normal subgroup of	#72287372, #72321460, #72321548, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391245);