#195779

order	512
nilpotency class	3
isoclinism	isoclinic class of #18524
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1074 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:

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

Polycyclic presentation:

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

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, 102975);