#193003

order	512
nilpotency class	3
isoclinism	isoclinic class of #18930
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

821 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fi,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, 100199);