#230425

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

534 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=eh,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,f^a=fi,d^b=de,e^b=eh,g^b=gh,d^c=dh,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, 137621);