#484142

order	512
nilpotency class	4
isoclinism	isoclinic class of #484139
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	#466	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1451 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

394 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)(19,30,33,40)(20,34)(21,27)(22,26)(23,38)(24,29)(25,31,42,48)(28,36)(32,45,39,46)(35,44)(37,41,47,43);(1,3)(2,7)(4,9)(5,11)(6,12)(8,10)(13,16)(14,15)(17,19,24,33)(18,25,29,42)(20,31,36,48)(21,32,23,39)(22,37,35,47)(26,40,44,30)(27,41,38,43)(28,45,34,46);(1,4,6,10,5,13,14,7)(2,3,9,12,8,11,16,15)(17,20,23,22,24,36,21,35)(18,26,27,34,29,44,38,28)(19,31,39,37,33,48,32,47)(25,40,41,46,42,30,43,45);(3,11)(4,13)(7,10)(12,15)(17,21,24,23)(18,27,29,38)(19,32,33,39)(20,35,36,22)(25,41,42,43)(26,34,44,28)(30,45,40,46)(31,47,48,37)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,d^2=i,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cg,e^a=eg,g^a=gi,d^b=dh,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	#72287372, #72321460, #72321472, …
maximal normal subgroup of	#72287372, #72321460, #72321472, …
maximal quotient of	#72343321, #72343429, #72343544, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391338);