#370953

order	512
nilpotency class	3
isoclinism	isoclinic class of #23912
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1254 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,d^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^b=fi,e^c=ei,f^c=fi,h^c=hi,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, 278149);