#438645

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

995 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)(3,4)(5,6,9,8)(7,11,12,10)(13,14)(15,25)(16,30)(17,23)(18,21)(19,24)(20,28)(22,32)(26,36)(27,35)(29,37)(31,43)(33,40)(34,38)(39,42)(41,44);(5,7)(6,10)(8,11)(9,12)(13,15)(14,20,24,37)(16,31,34,42)(17,28)(18,29)(19,26)(21,25,23,36)(22,39)(27,33,41,32)(30,35)(38,44)(40,43);(5,8,9,6)(7,11,12,10)(14,21)(15,26)(17,18)(20,25)(22,38)(23,24)(27,41)(30,40)(32,33)(35,39)(36,37)(43,44);(2,4)(7,12)(10,11)(13,16)(14,22)(15,27)(17,33)(18,32)(19,34)(20,35)(21,38)(23,30)(24,40)(25,39)(26,41)(28,31)(29,42)(36,43)(37,44)

Polycyclic presentation:

2^9|a^2=ef,b^2=gi,c^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dei,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=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, 345841);