#327071

order	512
nilpotency class	3
isoclinism	isoclinic class of #327021
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	C₂×C₄²	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1045 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)(11,19)(12,15)(13,17)(14,23)(16,21)(18,22)(20,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,39)(36,38)(37,40);(2,5)(6,8)(9,11)(10,15)(12,13)(14,20)(16,23)(17,19)(18,24)(21,22)(25,27,30,36)(26,31,34,38)(28,37,29,35)(32,40,33,39);(1,3)(2,6)(4,7)(5,8)(9,12,13,11)(10,16,17,24)(14,20,22,21)(15,18,19,23)(26,32)(27,35)(28,29)(33,34)(36,37)(39,40);(3,7)(6,8)(11,12)(15,19)(16,24)(20,21)(27,36)(31,38)(35,37)(39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 234267);