#210410

order	512
nilpotency class	3
isoclinism	isoclinic class of #19072
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

768 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)(7,12)(10,13)(15,16)(17,18,23,28,24,29,22,43)(19,30,32,25,34,39,33,44)(20,31,36,26,37,38,35,45)(21,27,41,46,42,47,40,48);(1,2,5,8)(3,6,10,15)(4,12,11,7)(9,16,13,14)(17,19)(18,25)(20,27)(21,38)(22,33)(23,32)(24,34)(26,40)(28,39)(29,44)(30,43)(31,42)(35,48)(36,46)(37,47)(41,45);(1,3)(2,6)(4,13)(5,10)(7,16)(8,15)(9,11)(12,14)(17,20)(18,26)(19,27)(21,39)(22,35)(23,36)(24,37)(25,40)(28,38)(29,45)(30,42)(31,43)(32,46)(33,48)(34,47)(41,44);(3,10)(6,15)(9,13)(14,16)(17,21,23,41,24,42,22,40)(18,27,28,46,29,47,43,48)(19,31,32,26,34,38,33,45)(20,30,36,25,37,39,35,44)

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=dh,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,f^c=fhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 117606);