#195420

order	512
nilpotency class	3
isoclinism	isoclinic class of #18376
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

1037 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)(19,29,32,39)(20,33)(21,27)(22,26)(23,28)(24,31,40,37)(25,35)(30,44,47,38)(34,42)(36,41)(43,46,48,45);(2,7)(4,5)(8,11)(9,10)(12,13)(15,16)(17,19,21,37,23,32,22,31)(18,24,27,39,28,40,26,29)(20,30,35,45,36,47,34,46)(25,38,41,48,42,44,33,43);(1,3,6,14)(2,8,10,16)(4,13,5,12)(7,11,9,15)(17,20,23,36)(18,25,28,42)(19,30,32,47)(21,35,22,34)(24,38,40,44)(26,33,27,41)(29,43,39,48)(31,46,37,45);(1,4,6,5)(2,9,10,7)(3,12,14,13)(8,11,16,15)(17,21,23,22)(18,26,28,27)(19,31,32,37)(20,34,36,35)(24,39,40,29)(25,41,42,33)(30,45,47,46)(38,43,44,48)

Polycyclic presentation:

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

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, 102616);