#483877

order	512
nilpotency class	4
isoclinism	isoclinic class of #483873
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 ◁ C₂×#322 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

432 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)(3,6,9,13,10,12,15,16)(17,18,22,26,23,28,20,25)(19,29,30,46,33,47,32,44)(21,35,38,45,37,43,34,27)(24,39,40,31,42,36,41,48);(17,19)(18,24)(20,30)(21,36)(22,32)(23,33)(25,40)(26,41)(27,44)(28,42)(29,43)(31,38)(34,48)(35,47)(37,39)(45,46);(1,3,4,9,5,10,11,15)(2,6,8,13,7,12,14,16)(18,25,28,26)(21,37)(24,40,42,41)(27,35,45,43)(29,44,47,46)(31,48)(34,38)(36,39);(2,7)(6,12)(8,14)(13,16)(17,20,23,22)(18,26,28,25)(19,30,33,32)(21,34,37,38)(24,41,42,40)(27,35,45,43)(29,44,47,46)(31,36,48,39)

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=dh,e^a=eg,g^a=gi,e^b=egi,f^b=fi,g^b=gi,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216779, #72216790, #72216898, …
maximal normal subgroup of	#72216779, #72216790, #72216898, …
maximal quotient of	C₂×#483877, #72318636, #72318691, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391073);