#345833

order	512
nilpotency class	3
isoclinism	isoclinic class of #345808
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3497	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

582 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=h,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fh,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=ghi,e^d=eh,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, 253029);