#378079

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

915 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=e,d^2=fh,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=di,f^c=fi,g^c=gi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 285275);