#390498

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

1039 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)(8,10)(11,12)(13,14,19,25)(15,20,28,21)(16,29,27,37)(17,32,35,40)(18,24,30,23)(22,34,39,33)(26,41,43,44)(31,38,42,36);(5,7,9,6)(8,11,12,10)(13,15)(14,20)(16,18)(17,26)(19,28)(21,25)(22,36)(23,37)(24,29)(27,30)(31,34)(32,41)(33,42)(35,43)(38,39)(40,44);(1,3,4,2)(5,8,9,12)(6,10,7,11)(13,16,30,28,19,27,18,15)(14,21,23,37,25,20,24,29)(17,31,33,43,35,42,34,26)(22,38,40,44,39,36,32,41);(6,7)(10,11)(13,17,19,35)(14,22,25,39)(15,26,28,43)(16,31,27,42)(18,34,30,33)(20,36,21,38)(23,40,24,32)(29,41,37,44)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=egh,d^2=i,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,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, 297694);