#195002

order	512
nilpotency class	3
isoclinism	isoclinic class of #18346
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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×#386
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1096 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)(3,11,13,16)(4,8)(5,10)(6,9)(7,12,15,14)(17,18,21,26,22,25,34,39)(19,23,29,37,28,36,43,47)(20,30,32,40,31,24,33,38)(27,35,42,46,41,45,44,48);(1,3,5,14,6,13,4,12)(2,7,10,16,9,15,8,11)(17,19,21,29,22,28,34,43)(18,23,26,37,25,36,39,47)(20,27,33,44,31,41,32,42)(24,35,40,48,30,45,38,46);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,11,15,16)(17,20)(18,24)(19,27)(21,33)(22,31)(23,35)(25,30)(26,40)(28,41)(29,44)(32,34)(36,45)(37,48)(38,39)(42,43)(46,47);(2,9)(3,13)(8,10)(12,14)(18,25)(23,36)(24,30)(26,39)(35,45)(37,47)(38,40)(46,48)

Polycyclic presentation:

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

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, 102198);