#484128

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	#33733	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

437 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fi,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=ci,d^a=di,e^a=eg,g^a=gi,d^b=dh,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, #72234326, #72234333, …
maximal normal subgroup of	#72234320, #72234326, #72234333, …
maximal quotient of	#72320193

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391324);