#193006

order	512
nilpotency class	3
isoclinism	isoclinic class of #18866
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	C₂×#121	Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

843 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)(3,12)(4,11)(5,10)(6,9)(7,14)(8,15)(13,16)(17,18,22,26,23,27,21,35)(19,28,31,24,30,39,36,38)(20,25,34,41,32,40,33,45)(29,42,43,37,44,47,46,48);(1,3)(2,7)(4,15)(5,14)(6,13)(8,9)(10,16)(11,12)(17,19,22,31,23,30,21,36)(18,24,26,39,27,38,35,28)(20,29,34,43,32,44,33,46)(25,37,41,47,40,48,45,42);(1,3)(2,8)(4,15)(5,14)(6,13)(7,9)(10,12)(11,16)(17,20)(18,25)(19,29)(21,33)(22,34)(23,32)(24,37)(26,41)(27,40)(28,42)(30,44)(31,43)(35,45)(36,46)(38,48)(39,47);(2,9)(3,13)(10,11)(14,15)(19,30)(20,32)(24,38)(25,40)(28,39)(31,36)(33,34)(41,45)

Polycyclic presentation:

2^9|a^2=g,b^2=g,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,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

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 100202);