#225500

order	512
nilpotency class	3
isoclinism	isoclinic class of #25440
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	D₄×C₄⋊C₄
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

610 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 132696);