#194623

order	512
nilpotency class	3
isoclinism	isoclinic class of #19014
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂×#386
prime radical	#60676	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1017 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=h,e^2=h,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fh,d^b=dh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fhi,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, 101819);