#294568

order	512
nilpotency class	3
isoclinism	isoclinic class of #272923
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#2656	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

945 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)(3,7)(4,5)(6,8)(9,10,13,17)(11,18,21,19)(12,23,14,16)(15,20,24,22)(25,26)(27,33)(28,31)(29,32)(30,35)(34,38)(36,40)(37,39);(2,5)(6,7)(9,10)(11,19)(12,16)(13,17)(14,23)(15,22)(18,21)(20,24)(25,27)(26,30)(28,36)(29,35)(31,38)(32,33)(34,37)(39,40);(9,11,14,20,13,21,12,22)(10,15,23,19,17,24,16,18)(25,28)(26,31)(27,34,35,36)(29,37)(30,38,33,40)(32,39);(1,3,4,8)(2,6,5,7)(9,12,13,14)(10,16,17,23)(11,20,21,22)(15,19,24,18)(26,32)(30,33)(31,39)(38,40)

Polycyclic presentation:

2^9|a^2=h,c^2=fh,d^2=ei,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,d^c=dh,g^c=ghi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 201764);