#320514

order	512
nilpotency class	3
isoclinism	isoclinic class of #298795
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

696 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)(3,7)(4,5)(6,8)(9,10,14,16,12,17,13,18)(11,19,22,23,20,15,21,24)(25,26)(27,30)(28,31)(29,32)(33,37)(34,40)(35,39)(36,38);(2,5)(6,7)(9,11)(10,15)(12,20)(13,21)(14,22)(16,24)(17,19)(18,23)(25,27,33,34,28,35,29,36)(26,30,37,40,31,39,32,38);(10,16)(11,20)(13,14)(15,23)(17,18)(19,24)(26,31)(27,34)(29,33)(30,38)(35,36)(39,40);(1,3,4,8)(2,6,5,7)(10,17)(15,19)(16,18)(23,24)(27,35)(30,39)(34,36)(38,40)

Polycyclic presentation:

2^9|a^2=fhi,b^2=ghi,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,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, 227710);