#210344

order	512
nilpotency class	3
isoclinism	isoclinic class of #18746
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#120
prime radical	#3035	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

861 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	SD₁₆×#444, #72342636, #72342642
maximal normal subgroup of	SD₁₆×#444, #72342636, #72342642

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 117540);