#231472

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
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	#2366
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

655 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 138668);