#369980

order	512
nilpotency class	3
isoclinism	isoclinic class of #23876
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1202 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^c=fi,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 277176);