#205935

order	512
nilpotency class	3
isoclinism	isoclinic class of #205761
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#390
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

862 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 113131);