#370366

order	512
nilpotency class	3
isoclinism	isoclinic class of #24384
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×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	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

834 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)(3,7)(4,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,26,22,27,21,37)(19,24,31,39,30,40,29,46)(20,32,36,42,34,48,35,25)(28,43,45,41,33,47,44,38);(2,5)(6,7)(9,11,12,15)(10,13,14,16)(17,19,21,29,22,30,23,31)(18,24,37,46,27,40,26,39)(20,33,35,45,34,28,36,44)(25,41,48,43,42,38,32,47);(10,14)(11,15)(17,20,23,36,22,34,21,35)(18,25,37,48,27,42,26,32)(19,28,31,45,30,33,29,44)(24,38,46,47,40,41,39,43);(1,3,4,8)(2,6,5,7)(20,34)(25,42)(28,33)(32,48)(35,36)(38,41)(43,47)(44,45)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=fg,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^c=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, 277562);