#303902

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

784 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)(9,10,15,18)(11,16,20,24)(12,14,17,13)(19,21,22,23)(25,26)(27,31)(28,38)(29,32)(30,33)(34,36)(35,39)(37,40);(1,3,5,4)(2,6,8,7)(9,11)(10,16)(12,21)(13,19)(14,22)(15,20)(17,23)(18,24)(25,27)(26,31)(28,34)(29,35)(30,37)(32,39)(33,40)(36,38);(1,3,5,4)(2,7,8,6)(9,12,13,10,15,17,14,18)(11,16,19,23,20,24,22,21)(25,28,38,26,30,32,29,33)(27,34,35,40,37,39,36,31);(1,4,5,3)(2,7,8,6)(9,13,15,14)(10,12,18,17)(11,19,20,22)(16,21,24,23)(25,29,30,38)(26,32,33,28)(27,35,37,36)(31,39,40,34)

Polycyclic presentation:

2^9|a^2=hi,b^2=e,c^2=efi,d^2=eh,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, 211098);