#265182

order	512
nilpotency class	3
isoclinism	isoclinic class of #265163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#547	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1008 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=i,d^2=eh,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,f^b=fhi,g^b=ghi,d^c=di,f^c=fi,g^c=ghi

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, 172378);