#348654

order	512
nilpotency class	3
isoclinism	isoclinic class of #348654
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18995	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

864 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)(10,15)(12,23)(13,27)(16,28)(18,30)(19,35)(20,21)(25,26)(29,34)(31,36)(32,33)(37,38);(9,10,13,15,14,18,27,30)(11,16,20,28,22,33,21,32)(12,24,25,34,26,17,23,29)(19,36,35,40,37,31,38,39);(1,2)(3,8)(4,6)(5,7)(9,11)(10,16)(12,19)(13,20)(14,22)(15,28)(17,31)(18,33)(21,27)(23,38)(24,36)(25,35)(26,37)(29,39)(30,32)(34,40);(3,7)(5,8)(9,12)(10,17)(11,19)(13,25)(14,26)(15,29)(16,31)(18,24)(20,35)(21,38)(22,37)(23,27)(28,39)(30,34)(32,40)(33,36)

Polycyclic presentation:

2^9|b^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,d^b=dh,f^c=fi,g^c=ghi,f^d=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, 255850);