#346001

order	512
nilpotency class	3
isoclinism	isoclinic class of #345968
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3205	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

544 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=e,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=def,f^a=fh,g^a=gh,c^b=cg,d^b=de,e^b=eh,f^b=fh,g^b=gi,f^c=fh,g^c=gi,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, 253197);