#230409

order	512
nilpotency class	3
isoclinism	isoclinic class of #19755
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

530 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:

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

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,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 137605);