#327128

order	512
nilpotency class	3
isoclinism	isoclinic class of #327117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#37621	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

970 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)(9,10,15,19)(11,16,12,21)(13,18,14,23)(17,20,24,22)(25,26)(27,30)(28,32)(29,31)(33,37)(34,39)(35,38)(36,40);(2,5)(6,8)(9,11)(10,16)(12,15)(13,22)(14,20)(17,18)(19,21)(23,24)(25,27)(26,30)(28,33)(29,35)(31,38)(32,37)(34,36)(39,40);(1,3)(2,6)(4,7)(5,8)(9,12,15,11)(10,17,19,24)(13,22,14,20)(16,23,21,18)(27,33)(28,36)(30,37)(32,40)(34,35)(38,39);(3,7)(6,8)(9,13,15,14)(10,18,19,23)(11,20,12,22)(16,24,21,17)(26,31)(30,38)(32,40)(37,39)

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,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

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 234324);