#327328

order	512
nilpotency class	3
isoclinism	isoclinic class of #326783
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	#2617	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1033 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)(3,6)(4,5)(7,8)(9,10)(11,18)(12,16)(13,21)(14,17)(15,19)(20,23)(22,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,39)(36,40)(37,38);(2,5)(6,8)(25,27)(26,31)(28,35)(29,37)(30,36)(32,39)(33,38)(34,40);(1,3)(2,6)(4,7)(5,8)(9,11,14,20)(10,15,17,24)(12,19,13,22)(16,23,21,18)(25,28,30,29)(26,32,34,33)(27,35,36,37)(31,38,40,39);(3,7)(6,8)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18)(25,29,30,28)(26,33,34,32)(27,35,36,37)(31,39,40,38)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 234524);