#229198

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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	#2361
prime radical	C₂×#41	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

589 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:

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

Polycyclic presentation:

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

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, 136394);