#374195

order	512
nilpotency class	3
isoclinism	isoclinic class of #196110
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1507 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281391);