#112389

order	512
nilpotency class	3
isoclinism	isoclinic class of #112377
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

798 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19585);