#197629

order	512
nilpotency class	3
isoclinism	isoclinic class of #19504
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	#60900	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1040 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,e^2=hi,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,d^b=dhi,e^b=ei,f^b=fi,g^b=gi,d^c=dh,e^c=ei,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, 104825);