#320304

order	512
nilpotency class	3
isoclinism	isoclinic class of #298711
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

672 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=ghi,c^2=i,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 227500);