#231538

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	#2366
prime radical	C₂³	#550
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

553 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(2,6,8,12)(7,9,14,15)(17,18,21,24)(19,25,26,32)(20,28,29,23)(22,30,31,27);(1,2,4,12,5,8,13,6)(3,9,10,7,11,15,16,14)(19,26)(22,31)(25,32)(27,30);(17,19,20,27,21,26,29,30)(18,22,23,25,24,31,28,32);(1,3,4,10,5,11,13,16)(2,7,6,9,8,14,12,15)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287409, #72287433, #72287441
maximal normal subgroup of	#72287409, #72287433, #72287441

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 138734);