#388500

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	#23386	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1316 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:

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

Polycyclic presentation:

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

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, 295696);