#370903

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

1070 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:

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

Polycyclic presentation:

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

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, 278099);