#379230

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 ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

961 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)(7,10)(8,9)(11,12)(13,14)(15,20)(16,27,31,40)(17,32)(18,23)(19,24)(21,30,39,34)(22,33)(25,38,42,44)(26,37)(28,43,29,36)(35,41);(1,2,3,4)(6,9)(10,12)(13,15,18,17,19,26,35,33)(14,20,23,32,24,37,41,22)(16,28,30,42,31,29,34,25)(21,38,27,43,39,44,40,36);(2,4)(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,25)(17,29)(18,34)(19,31)(20,36)(22,38)(23,40)(24,39)(26,42)(27,41)(28,33)(30,35)(32,44)(37,43);(6,9)(10,12)(13,17,18,26,19,33,35,15)(14,22,23,20,24,32,41,37)(16,29,34,42,31,28,30,25)(21,38,40,36,39,44,27,43)

Polycyclic presentation:

2^9|a^2=h,b^2=g,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, 286426);