#482221

order	512
nilpotency class	4
isoclinism	isoclinic class of #11858
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#102
prime radical	#60673	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

777 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)(3,9,11,6)(4,12)(5,8)(7,13)(10,14,16,15)(17,18,20,25,23,28,22,27)(19,29,32,43,33,46,30,45)(21,35,34,26,38,42,37,44)(24,36,40,31,41,47,39,48);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19)(18,24)(20,30)(21,36)(22,32)(23,33)(25,39)(26,43)(27,40)(28,41)(29,42)(31,37)(34,48)(35,46)(38,47)(44,45);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15);(17,20,23,22)(18,25,28,27)(19,30,33,32)(21,34,38,37)(24,39,41,40)(26,42,44,35)(29,45,46,43)(31,36,48,47)

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=i,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cgh,e^a=eg,g^a=gi,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,h^b=hi

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, 389417);