#231528

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

551 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)(3,9)(7,14,15,10)(11,16)(17,18,21,24)(19,25,28,32)(20,29,26,23)(22,30,31,27);(1,2,4,12,5,8,13,6)(3,10,11,7,9,14,16,15);(17,19,26,30,21,28,20,27)(18,22,29,32,24,31,23,25);(1,3,4,11,5,9,13,16)(2,7,6,10,8,15,12,14)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 138724);