#481214

order	512
nilpotency class	3
isoclinism	isoclinic class of #17597
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×C₂²⋊C₈
prime radical	#3475	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

714 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)(3,10)(4,9)(5,13)(6,8)(7,11)(12,16)(14,15)(17,18,21,27,23,28,36,43,22,24,35,37,20,26,34,42)(19,29,31,44,30,41,45,48,32,25,38,40,33,39,46,47);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(18,24)(21,35)(25,29)(26,28)(31,38)(34,36)(39,41)(45,46);(2,8)(7,15)(9,13)(10,16)(17,19)(18,25)(20,33)(21,31)(22,32)(23,30)(24,29)(26,41)(27,40)(28,39)(34,46)(35,38)(36,45)(37,44)(42,48)(43,47);(1,4,6,5)(2,9,8,13)(3,11,12,14)(7,16,15,10)(17,20,22,23)(18,26,24,28)(19,30,32,33)(21,34,35,36)(25,39,29,41)(27,42,37,43)(31,45,38,46)(40,47,44,48)

Polycyclic presentation:

2^9|a^2=f,d^2=g,e^2=gi,f^2=h,h^2=i,b^a=be,c^a=cg,e^a=eg,e^b=egi,f^b=fi,d^c=di

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388410);