#199213

order	512
nilpotency class	3
isoclinism	isoclinic class of #199212
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3036	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1499 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)(3,6)(7,8)(9,10,13,16)(11,14)(12,15,20,17)(18,22)(19,21)(23,24);(1,2,6,8,4,7,3,5)(9,11)(10,14,16,22)(12,19)(13,18)(15,21,17,23)(20,24);(10,15)(12,20)(14,21)(16,17)(19,24)(22,23);(10,16)(14,22)(15,17)(21,23)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,e^2=hi,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,e^b=ei,g^b=gi,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257594, #72316963, #72316965
maximal normal subgroup of	#72257594, #72316963, #72316965
maximal quotient of	#72257670

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 106409);