#364848

order	512
nilpotency class	3
isoclinism	isoclinic class of #196134
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1290 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=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, 272044);