#383811

order	512
nilpotency class	3
isoclinism	isoclinic class of #30484
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

1031 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,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

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 291007);