#195061

order	512
nilpotency class	3
isoclinism	isoclinic class of #18446
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	#36687	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1156 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=h,e^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dhi,f^a=fi,e^b=eh,f^b=fi,g^b=gh,d^c=dh,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, 102257);