#386723

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×C₄⋊C₄
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1078 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)(3,4)(5,7)(6,8)(9,10,15,20,13,19,14,17)(11,21,25,33,24,16,22,32)(12,26,30,18,28,36,29,35)(23,38,39,34,27,40,37,31);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,27)(13,24)(14,22)(15,25)(17,33)(18,34)(19,21)(20,32)(23,28)(26,38)(29,39)(30,37)(31,35)(36,40);(1,4,5,6)(2,3,7,8)(10,17,19,20)(11,22,24,25)(16,21)(18,26,35,36)(23,37,27,39)(31,34)(32,33)(38,40);(2,7)(3,8)(9,12,14,29,13,28,15,30)(10,18,20,36,19,35,17,26)(11,23,22,37,24,27,25,39)(16,31,32,38,21,34,33,40)

Polycyclic presentation:

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

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, 293919);