#265373

order	512
nilpotency class	3
isoclinism	isoclinic class of #265359
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#18229	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1049 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)(3,7)(4,6)(5,8)(9,10)(11,17)(12,25)(13,28)(14,21)(15,20)(16,23)(18,29)(19,26)(22,37)(24,32)(27,31)(30,36)(33,38)(34,39)(35,40);(1,3,4,8)(2,5,6,7)(9,11,14,23,15,12,26,24)(10,16,19,17,20,32,21,25)(13,22,29,38,30,27,40,39)(18,31,28,33,35,37,36,34);(5,7)(9,12)(10,17)(11,15)(13,22)(14,24)(16,21)(18,34)(19,32)(20,25)(23,26)(27,30)(28,31)(29,38)(33,35)(36,37)(39,40);(2,6)(5,7)(9,13,26,40,15,30,14,29)(10,18,21,36,20,35,19,28)(11,22,24,39,12,27,23,38)(16,31,25,34,32,37,17,33)

Polycyclic presentation:

2^9|b^2=e,d^2=ehi,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,d^c=dh,g^c=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 172569);