#376977

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1507 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=efi,d^2=h,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,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, 284173);