#439174

order	512
nilpotency class	3
isoclinism	isoclinic class of #417784
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

891 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:

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

Polycyclic presentation:

2^9|b^2=fg,c^2=ef,d^2=ei,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^c=gi,h^d=hi

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, 346370);