#375771

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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

1705 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,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, 282967);