#376974

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

1521 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)(9,10,14,19)(11,20,24,32)(12,25,28,34)(13,16,29,18)(15,21,30,23)(17,27,33,26)(22,31,38,39)(35,37,40,36);(9,11)(10,15)(12,22)(13,21)(14,24)(16,20)(17,31)(18,32)(19,30)(23,29)(25,35)(26,37)(27,36)(28,38)(33,39)(34,40);(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,c^2=efi,d^2=hi,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, 284170);