#192290

order	512
nilpotency class	3
isoclinism	isoclinic class of #18414
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

796 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^2=h,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=eh,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 99486);