#275000

order	512
nilpotency class	3
isoclinism	isoclinic class of #268763
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

657 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)(3,6)(4,5)(7,8)(9,10,14,17,13,18,12,22)(11,19,20,24,21,15,23,16)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(2,5)(6,8)(9,11)(10,15)(12,23)(13,21)(14,20)(16,17)(18,19)(22,24)(25,27,28,35)(26,30,31,33)(29,34,37,36)(32,38,40,39);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,20)(13,21)(14,23)(15,17)(18,24)(19,22)(27,34)(29,37)(30,38)(32,40)(33,39)(35,36);(2,5)(6,8)(9,12,13,14)(10,17,18,22)(11,20,21,23)(15,24,19,16)(26,31)(30,33)(32,40)(38,39)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=ehi,d^2=h,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, 182196);