#370462

order	512
nilpotency class	3
isoclinism	isoclinic class of #24390
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×#97 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

782 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)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,27,22,26,28,25)(19,23,31,39,30,37,43,38)(20,32,34,24,35,42,44,41)(29,45,46,36,33,47,48,40);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16)(17,19,28,43,22,30,21,31)(18,23,25,38,26,37,27,39)(20,33,44,46,35,29,34,48)(24,40,32,47,41,36,42,45);(1,3,5,4)(2,7,8,6)(10,14)(11,15)(17,20,28,44,22,35,21,34)(18,24,27,42,26,41,25,32)(19,29,43,48,30,33,31,46)(23,36,39,47,37,40,38,45);(1,4,5,3)(2,7,8,6)(19,30)(23,37)(29,33)(31,43)(36,40)(38,39)(45,47)(46,48)

Polycyclic presentation:

2^9|a^2=h,b^2=egh,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, 277658);