#297209

order	512
nilpotency class	3
isoclinism	isoclinic class of #268211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

654 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=h,c^2=fh,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=dhi,g^c=gh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 204405);