#303790

order	512
nilpotency class	3
isoclinism	isoclinic class of #271131
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

864 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)(4,6)(5,8)(9,10,14,18)(11,15,16,24)(12,21,22,20)(13,19,23,17)(25,26,30,35)(27,33,38,32)(28,39,37,40)(29,36,31,34);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,19)(13,20)(14,16)(17,22)(18,24)(21,23)(25,27,30,38)(26,32,35,33)(28,31,37,29)(34,39,36,40);(1,3,5,4)(2,7,8,6)(9,12,13,15,14,22,23,24)(10,16,17,21,18,11,19,20)(25,28,31,38,30,37,29,27)(26,33,36,40,35,32,34,39);(1,4,5,3)(2,7,8,6)(11,16)(12,22)(15,24)(20,21)(25,29,30,31)(26,34,35,36)(27,37,38,28)(32,40,33,39)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=ef,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,d^c=dhi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 210986);