#484019

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	#33666	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

439 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)(17,18,21,27,24,28,23,39)(19,29,31,45,33,46,40,41)(20,34,35,48,36,26,22,37)(25,38,42,30,44,47,43,32);(17,19)(18,25)(20,30)(21,31)(22,38)(23,40)(24,33)(26,41)(27,42)(28,44)(29,37)(32,36)(34,45)(35,47)(39,43)(46,48);(1,3,4,10,5,9,11,15)(2,6,7,13,8,12,14,16)(17,20,23,22,24,36,21,35)(18,26,39,48,28,34,27,37)(19,30,40,38,33,32,31,47)(25,41,43,46,44,45,42,29);(3,9)(6,12)(10,15)(13,16)(17,21,24,23)(18,27,28,39)(19,31,33,40)(20,35,36,22)(25,42,44,43)(26,37,34,48)(29,45,46,41)(30,47,32,38)

Polycyclic presentation:

2^9|a^2=f,c^2=fi,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=ci,e^a=eg,g^a=gi,e^b=eg,g^b=gi,d^c=dhi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234320, #72260370, #72260406, …
maximal normal subgroup of	#72234320, #72260370, #72260406, …
maximal quotient of	#72320199

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391215);