#379320

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

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

853 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,8)(4,6)(5,7)(9,10)(11,16)(12,15)(13,14)(17,18,24,30,22,28,23,29)(19,31,21,38,33,25,34,27)(20,26,37,43,36,42,39,44)(32,45,35,48,46,40,47,41);(1,3,5,4)(2,6,7,8)(10,14)(15,16)(17,19,24,21,22,33,23,34)(18,25,30,27,28,31,29,38)(20,35,39,32,36,47,37,46)(26,41,44,40,42,48,43,45);(1,4,5,3)(2,6,7,8)(9,11,13,12)(10,15,14,16)(17,20,22,36)(18,26,28,42)(19,32,33,46)(21,35,34,47)(23,39,24,37)(25,40,31,45)(27,41,38,48)(29,44,30,43);(2,7)(6,8)(9,12,13,11)(10,16,14,15)(17,21,24,33,22,34,23,19)(18,27,30,31,28,38,29,25)(20,35,37,46,36,47,39,32)(26,41,43,45,42,48,44,40)

Polycyclic presentation:

2^9|a^2=h,b^2=efg,c^2=e,d^2=fh,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,g^c=gi

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, 286516);