#370875

order	512
nilpotency class	3
isoclinism	isoclinic class of #23912
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

1064 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:

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

Polycyclic presentation:

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

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, 278071);