#375689

order	512
nilpotency class	3
isoclinism	isoclinic class of #213778
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1255 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=fhi,d^2=eg,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,h^b=hi,e^c=ei,g^c=gi,f^d=fi,h^d=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, 282885);