#388620

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

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

1040 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2,4,3)(5,6)(7,9)(8,11)(10,12)(13,14,19,24)(15,20,28,36)(16,29,31,37)(17,21,18,23)(22,32,38,30)(25,34,27,35)(26,40,42,44)(33,41,43,39);(5,7)(6,8)(9,12)(10,11)(13,15)(14,20)(16,26)(17,27)(18,25)(19,28)(21,35)(22,33)(23,34)(24,36)(29,40)(30,39)(31,42)(32,41)(37,44)(38,43);(1,3,4,2)(5,8,10,9)(6,11,12,7)(14,21)(15,25)(17,18)(22,29)(23,24)(26,39)(27,28)(30,32)(34,35)(37,38)(40,44)(41,42);(5,9,10,8)(6,11,12,7)(13,16,19,31)(14,22,24,38)(15,26,28,42)(17,32,18,30)(20,33,36,43)(21,29,23,37)(25,39,27,41)(34,44,35,40)

Polycyclic presentation:

2^9|a^2=h,c^2=ehi,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, 295816);