#357455

order	512
nilpotency class	3
isoclinism	isoclinic class of #23400
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4406 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)(5,6,7,8)(9,10)(11,20)(12,17)(13,18)(14,27)(15,19)(16,23)(21,36)(22,30)(24,35)(25,32)(26,34)(28,33)(29,39)(31,37)(38,40);(9,11,15,24)(10,16,19,30)(12,21,26,38)(13,23,14,22)(17,31,34,29)(18,35,27,20)(25,39,28,37)(32,36,33,40);(2,4)(9,12)(10,17)(11,21)(13,25)(14,28)(15,26)(16,29)(18,32)(19,34)(20,36)(22,37)(23,39)(24,38)(27,33)(30,31)(35,40);(9,13,15,14)(10,18,19,27)(11,22,24,23)(12,25,26,28)(16,20,30,35)(17,32,34,33)(21,37,38,39)(29,36,31,40)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=i,e^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,d^b=di,e^b=ei,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314500, #72314513, #72314533, …
maximal normal subgroup of	#72314500, #72314513, #72314533, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264651);