#275550

order	512
nilpotency class	3
isoclinism	isoclinic class of #268763
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

528 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=e,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,g^b=gi,f^c=fh,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 182746);