#227173

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2358
prime radical	C₂×#454	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

747 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,5,7,4,3,6,8)(9,10,16,22)(11,23,12,26)(13,19,31,37)(14,21,15,20)(17,28,18,25)(24,39,27,35)(29,34,40,33)(30,38,32,36);(1,3,5,8,4,2,6,7)(9,11,16,12)(10,17,22,18)(13,29,31,40)(14,28,15,25)(19,35,37,39)(20,26,21,23)(24,32,27,30)(33,38,34,36);(2,3)(7,8)(9,12,16,11)(10,18,22,17)(13,24,31,27)(14,28,15,25)(19,33,37,34)(20,26,21,23)(29,32,40,30)(35,38,39,36);(9,13,16,31)(10,19,22,37)(11,24,12,27)(14,32,15,30)(17,33,18,34)(20,38,21,36)(23,39,26,35)(25,40,28,29)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,c^b=ci,d^b=de,e^b=eh,d^c=dh,f^c=fh,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, 134369);