#401108

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2408
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1239 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 308304);