#192951

order	512
nilpotency class	3
isoclinism	isoclinic class of #18810
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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

839 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)(4,12)(5,8)(6,11)(9,14)(10,16)(13,15)(17,18,22,27,23,26,21,34)(19,24,29,40,30,38,35,39)(20,31,33,25,32,42,36,41)(28,43,46,48,44,47,45,37);(1,3)(2,6)(4,13)(5,10)(7,12)(8,15)(9,16)(11,14)(17,19,22,29,23,30,21,35)(18,24,27,40,26,38,34,39)(20,28,33,46,32,44,36,45)(25,37,42,43,41,48,31,47);(1,3)(2,7)(4,11)(5,10)(6,12)(8,16)(9,15)(13,14)(17,20)(18,25)(19,28)(21,36)(22,33)(23,32)(24,37)(26,41)(27,42)(29,46)(30,44)(31,34)(35,45)(38,48)(39,47)(40,43);(2,8)(3,10)(9,12)(11,13)(18,26)(24,38)(25,41)(27,34)(31,42)(37,48)(39,40)(43,47)

Polycyclic presentation:

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

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, 100147);