#304988

order	512
nilpotency class	3
isoclinism	isoclinic class of #269211
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	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

677 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=eh,c^2=ef,d^2=ehi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dh,g^c=gh

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, 212184);