#194469

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	C₂×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1026 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)(3,12,14,15)(4,13,16,8)(5,9)(6,10)(7,11)(17,18)(19,28,30,39)(20,32)(21,27)(22,36)(23,26)(24,31,40,35)(25,34)(29,44,47,38)(33,41)(37,42)(43,45,48,46);(1,3,5,13,7,14,6,8)(2,4,9,12,11,16,10,15)(17,19,21,35,23,30,22,31)(18,24,27,39,26,40,36,28)(20,29,34,46,33,47,37,45)(25,38,41,48,42,44,32,43);(1,4)(2,8)(3,9)(5,12)(6,15)(7,16)(10,14)(11,13)(17,20)(18,25)(19,29)(21,34)(22,37)(23,33)(24,38)(26,42)(27,41)(28,43)(30,47)(31,45)(32,36)(35,46)(39,48)(40,44);(1,5,7,6)(2,9,11,10)(3,8,14,13)(4,15,16,12)(18,26)(19,30)(20,33)(27,36)(31,35)(34,37)(38,44)(43,48)

Polycyclic presentation:

2^9|a^2=g,b^2=egi,d^2=i,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=dhi,f^a=fh,d^b=dh,e^b=eh,f^b=fi,g^b=gh,d^c=dh,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, 101665);