#195392

order	512
nilpotency class	3
isoclinism	isoclinic class of #18284
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	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1030 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102588);