#483966

order	512
nilpotency class	4
isoclinism	isoclinic class of #483962
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 ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

401 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fg,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cg,d^a=dhi,e^a=eg,g^a=gi,d^b=dh,e^b=egi,f^b=fi,g^b=gi,d^c=di,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321740, #72321751, #72321763, …
maximal normal subgroup of	#72321740, #72321751, #72321763, …
maximal quotient of	#72343318, #72343426, #72343547, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391162);