#260695

order	512
nilpotency class	3
isoclinism	isoclinic class of #247367
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2541	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

823 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,20)(13,22)(14,18)(16,21)(17,23)(19,24)(25,26)(27,31)(28,33)(29,32)(30,34)(35,38)(36,40)(37,39);(2,5)(6,7)(9,11)(10,15)(12,14)(13,19)(16,17)(18,20)(21,23)(22,24)(25,27)(26,31)(28,36)(29,35)(30,37)(32,38)(33,40)(34,39);(9,12,14,11)(10,16,18,24)(13,19,23,21)(15,22,20,17)(26,32)(27,35)(28,29)(33,34)(36,37)(38,40);(1,3,4,8)(2,6,5,7)(11,12)(15,20)(16,24)(19,21)(25,28,30,29)(26,33,34,32)(27,35,37,36)(31,38,39,40)

Polycyclic presentation:

2^9|c^2=hi,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,d^c=dh,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, 167891);