#369391

order	512
nilpotency class	3
isoclinism	isoclinic class of #24390
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1438 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)(7,11)(8,10)(9,12)(13,14,17,23,18,22,29,21)(15,19,26,35,25,33,41,34)(16,27,30,20,31,38,40,37)(24,39,42,32,28,43,44,36);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15,17,26,18,25,29,41)(14,19,23,35,22,33,21,34)(16,28,30,44,31,24,40,42)(20,36,38,39,37,32,27,43);(2,4)(5,7,9,8)(6,11,12,10)(13,16,29,40,18,31,17,30)(14,20,23,38,22,37,21,27)(15,24,41,44,25,28,26,42)(19,32,35,43,33,36,34,39);(5,8,9,7)(6,11,12,10)(15,25)(19,33)(24,28)(26,41)(32,36)(34,35)(39,43)(42,44)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=efg,d^2=e,f^2=i,h^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, 276587);