#193150

order	512
nilpotency class	3
isoclinism	isoclinic class of #18992
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂⁴	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

777 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 100346);