#483926

order	512
nilpotency class	4
isoclinism	isoclinic class of #483919
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 ◁ C₂×#322 ◁ 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:

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

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286569, #72286677, #72286959, …
maximal normal subgroup of	#72286569, #72286677, #72286959, …
maximal quotient of	#72343707, #72343765

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391122);