#295208

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	#2819	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

797 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)(9,10,15,19)(11,16,20,17)(12,21,22,24)(13,23,14,18)(25,26,28,30)(27,33,34,40)(29,37,35,32)(31,39,38,36);(2,5)(6,7)(9,11)(10,16)(12,13)(14,22)(15,20)(17,19)(18,24)(21,23)(26,30)(27,34)(29,35)(31,38);(9,12,14,11,15,22,13,20)(10,17,23,24,19,16,18,21)(25,27,35,39,28,34,29,36)(26,31,37,40,30,38,32,33);(1,3,4,8)(2,6,5,7)(26,30)(27,34)(32,37)(36,39)

Polycyclic presentation:

2^9|a^2=h,c^2=fh,d^2=ei,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=di,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, 202404);