#383605

order	512
nilpotency class	3
isoclinism	isoclinic class of #30490
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	D₄×C₂²⋊C₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1044 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 290801);