#229602

order	512
nilpotency class	3
isoclinism	isoclinic class of #19883
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2361
prime radical	#3035	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

607 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=e,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,f^a=fi,c^b=ci,d^b=de,e^b=eh,g^b=gh,f^c=fi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	SD₁₆×#444, #72342634, #72342647
maximal normal subgroup of	SD₁₆×#444, #72342634, #72342647

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 136798);