#192539

order	512
nilpotency class	3
isoclinism	isoclinic class of #18224
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	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

800 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:

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

Polycyclic presentation:

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

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, 99735);