#347461

order	512
nilpotency class	3
isoclinism	isoclinic class of #347434
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#63052	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2197 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=h,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fhi,g^a=gh,e^b=eh,f^b=fi,e^c=ei,f^c=fhi,g^d=gh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254657);