#354111

order	512
nilpotency class	3
isoclinism	isoclinic class of #3068
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2778 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,e^b=ei,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261307);