#347458

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	#63066	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2254 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)(11,23,28,36)(12,29,25,35)(13,32,24,40)(14,21,15,20)(17,27,37,26)(18,30,38,31)(19,34,39,33);(3,7)(5,8)(9,11)(10,17)(12,24)(13,25)(14,27)(15,26)(16,28)(18,32)(19,35)(20,36)(21,23)(22,37)(29,39)(30,33)(31,34)(38,40);(1,2)(3,8)(4,6)(5,7)(9,12,16,25)(10,18,22,38)(11,24,28,13)(14,30,15,31)(17,32,37,40)(19,23,39,36)(20,35,21,29)(26,34,27,33);(9,13,16,24)(10,19,22,39)(11,25,28,12)(14,33,15,34)(17,35,37,29)(18,23,38,36)(20,32,21,40)(26,31,27,30)

Polycyclic presentation:

2^9|a^2=h,c^2=h,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, 254654);