#210403

order	512
nilpotency class	3
isoclinism	isoclinic class of #19014
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	#3531	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

747 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=gi,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,e^c=ei,f^c=fhi

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, 117599);