#482669

order	512
nilpotency class	4
isoclinism	isoclinic class of #482367
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

924 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

(2,6)(3,4)(7,8)(9,10,14,17,15,19,11,20)(12,16,13,18,21,24,22,23);(1,2)(3,6)(4,8)(5,7)(9,11,15,14)(10,16,19,24)(12,22,21,13)(17,23,20,18);(1,3,5,4)(2,6,7,8)(9,11,15,14)(10,17,19,20)(12,13,21,22)(16,23,24,18);(1,3,5,4)(2,6,7,8)(9,12)(10,16)(11,13)(14,22)(15,21)(17,23)(18,20)(19,24)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=ch,d^a=dg,e^a=eg,g^a=gi,d^b=di,e^b=eh,f^b=fgh,f^d=fi,f^e=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389865);