#205726

order	512
nilpotency class	3
isoclinism	isoclinic class of #205569
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	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

900 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)(3,9)(4,13)(5,7)(6,12)(8,10)(11,15)(14,16)(17,18,22,27,23,26,21,35)(19,24,30,40,29,38,36,39)(20,31,34,41,32,45,33,25)(28,42,44,37,43,47,46,48);(1,3,10,16,5,11,4,12)(2,6,13,15,7,14,8,9)(17,19,21,30,23,29,22,36)(18,24,27,39,26,38,35,40)(20,28,33,44,32,43,34,46)(25,37,31,42,41,48,45,47);(2,7)(6,14)(8,13)(9,15)(17,20,22,34,23,32,21,33)(18,25,35,45,26,41,27,31)(19,28,36,46,29,43,30,44)(24,37,40,47,38,48,39,42);(3,11)(6,14)(9,15)(12,16)(19,29)(20,32)(24,38)(25,41)(30,36)(31,45)(33,34)(39,40)

Polycyclic presentation:

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

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, 112922);