#226956

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×H(5,𝔽₂)	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

759 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dei,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, 134152);