#369076

order	512
nilpotency class	3
isoclinism	isoclinic class of #24390
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1902 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=fgi,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di

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, 276272);