#363426

order	512
nilpotency class	3
isoclinism	isoclinic class of #214672
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1278 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)(4,5)(6,8)(9,10)(11,16,23,33)(12,25,13,18)(14,20)(15,19)(17,26,36,27)(21,38)(22,31)(24,32,29,37)(28,35)(30,40)(34,39);(2,5)(6,7)(9,11)(10,16)(12,21)(13,22)(14,23)(15,29)(17,34)(18,31)(19,37)(20,33)(24,28)(25,38)(26,40)(27,39)(30,36)(32,35);(9,12,14,13)(10,17,20,36)(11,21,23,22)(15,27,28,26)(16,30,33,34)(18,35,25,19)(24,40,29,39)(31,37,38,32);(1,3,4,8)(2,6,5,7)(9,13,14,12)(10,18,20,25)(11,22,23,21)(15,27,28,26)(16,31,33,38)(17,35,36,19)(24,40,29,39)(30,37,34,32)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=eg,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,h^b=hi,f^c=fi,h^c=hi,h^d=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, 270622);