#267910

order	512
nilpotency class	3
isoclinism	isoclinic class of #267899
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1012 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:

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

Polycyclic presentation:

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

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, 175106);