#230416

order	512
nilpotency class	3
isoclinism	isoclinic class of #20022
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2361
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

526 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=e,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dhi,f^a=fi,d^b=de,e^b=eh,g^b=gh,f^c=fi

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, 137612);