#118123

order	512
nilpotency class	3
isoclinism	isoclinic class of #10637
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#397
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂²×#41	Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

601 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,e^b=ei,d^c=di,f^c=fi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25319);