#220228

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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×C₂²⋊C₄
prime radical	C₂×#578	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1023 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 127424);