#228157

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	#2358
prime radical	#18520	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

680 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:

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

Polycyclic presentation:

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

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, 135353);