#210418

order	512
nilpotency class	3
isoclinism	isoclinic class of #19086
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#120
prime radical	#18990	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

749 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=g,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fhi,d^b=di,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^c=fhi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 117614);