#383511

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1400 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,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, 290707);