#205628

order	512
nilpotency class	3
isoclinism	isoclinic class of #205557
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

885 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)(3,11,12,16,13,8,14,15)(17,18)(19,28,32,39)(20,33)(21,26)(22,37)(23,27)(24,31,40,30)(25,34)(29,38,45,48)(35,42)(36,41)(43,46,47,44);(2,7,10,9)(8,15,11,16)(17,19,21,30,23,32,22,31)(18,24,26,39,27,40,37,28)(20,29,35,46,36,45,34,44)(25,38,33,47,42,48,41,43);(1,3)(2,8)(4,14)(5,12)(6,13)(7,15)(9,16)(10,11)(17,20,22,34,23,36,21,35)(18,25,26,33,27,42,37,41)(19,29,31,44,32,45,30,46)(24,38,39,47,40,48,28,43);(1,4,6,5)(2,9,10,7)(3,12,13,14)(8,15,11,16)(17,21,23,22)(18,26,27,37)(19,30,32,31)(20,34,36,35)(24,39,40,28)(25,41,42,33)(29,44,45,46)(38,43,48,47)

Polycyclic presentation:

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

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, 112824);