#205704

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	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

856 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)(4,7)(8,12)(9,13)(11,14)(15,16)(17,18,22,28,23,27,21,26)(19,24,32,39,31,40,30,38)(20,33,34,41,36,42,35,25)(29,43,44,37,45,47,46,48);(1,2,7,13,5,9,4,6)(3,8,14,16,10,15,11,12)(17,19,21,32,23,31,22,30)(18,24,28,38,27,40,26,39)(20,29,35,44,36,45,34,46)(25,37,33,43,41,48,42,47);(1,3,5,10)(2,8,9,15)(4,11,7,14)(6,12,13,16)(17,20,22,34,23,36,21,35)(18,25,26,42,27,41,28,33)(19,29,30,46,31,45,32,44)(24,37,39,47,40,48,38,43);(2,9)(3,10)(6,13)(11,14)(17,21,23,22)(18,26,27,28)(19,30,31,32)(20,34,36,35)(24,38,40,39)(25,33,41,42)(29,44,45,46)(37,47,48,43)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=fh,d^2=i,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=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, 112900);