#357675

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₄⋊C₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2678 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)(9,10,14,18,15,19,13,28)(11,20,24,30,22,34,23,16)(12,17,27,32,25,31,26,38)(21,33,36,39,37,40,35,29);(5,6)(7,8)(9,11,13,23,15,22,14,24)(10,16,18,20,19,30,28,34)(12,21,26,35,25,37,27,36)(17,29,32,33,31,39,38,40);(2,4)(6,8)(9,12)(10,17)(11,21)(13,26)(14,27)(15,25)(16,29)(18,32)(19,31)(20,33)(22,37)(23,35)(24,36)(28,38)(30,39)(34,40);(11,22)(12,25)(16,30)(17,31)(20,34)(23,24)(26,27)(32,38)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264871);