#229221

order	512
nilpotency class	3
isoclinism	isoclinic class of #19755
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	C₂³	#548
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

527 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)(19,25,27,32)(20,28,30,23)(22,29,31,26);(1,2,4,12,5,8,13,6)(3,9,11,7,10,15,16,14);(3,10)(7,14)(9,15)(11,16)(17,19,21,27)(18,22,24,31)(20,29,30,26)(23,32,28,25);(1,3,4,11,5,10,13,16)(2,7,6,9,8,14,12,15)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 136417);