#484052

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

412 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)(19,31,34,40)(20,35)(21,29)(22,26)(23,27)(24,30)(25,32,42,48)(28,37)(33,44,39,46)(36,43)(38,45,47,41);(17,19)(18,25)(20,32)(21,33)(22,38)(23,39)(24,34)(26,40)(27,41)(28,44)(29,45)(30,42)(31,43)(35,46)(36,47)(37,48);(1,3,4,10,5,9,11,15)(2,6,7,13,8,12,14,16)(17,20,23,22,24,37,21,36)(18,26,29,35,30,43,27,28)(19,32,39,38,34,48,33,47)(25,40,45,46,42,31,41,44);(3,9)(6,12)(10,15)(13,16)(17,21,24,23)(18,27,30,29)(19,33,34,39)(20,36,37,22)(25,41,42,45)(26,28,43,35)(31,46,40,44)(32,47,48,38)

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,d^a=di,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	#72286644, #72286648, #72286653, …
maximal normal subgroup of	#72286644, #72286648, #72286653, …
maximal quotient of	#72343291, #72343442, #72343498, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391248);