#497503

order	512
nilpotency class	4
isoclinism	isoclinic class of #480872
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	C₂²×Q₈	C₂⁴
prime radical	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#548 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1011 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=de,f^c=fi,g^c=ghi,g^f=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404699);