#205661

order	512
nilpotency class	3
isoclinism	isoclinic class of #9758
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#390
prime radical	C₂⁴	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

908 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)(4,12)(5,7)(6,11)(8,13)(10,15)(14,16)(17,18,22,28,23,27,21,26)(19,24,30,39,31,41,37,40)(20,32,35,44,36,43,34,25)(29,45,46,38,33,47,48,42);(1,3,4,11,5,10,13,16)(2,6,8,9,7,14,12,15)(17,19,22,37,23,31,21,30)(18,24,26,39,27,41,28,40)(20,33,35,46,36,29,34,48)(25,42,43,45,44,38,32,47);(2,7)(3,10)(8,12)(11,16)(17,20,22,35,23,36,21,34)(18,25,26,43,27,44,28,32)(19,29,37,48,31,33,30,46)(24,38,39,47,41,42,40,45);(17,21,23,22)(18,26,27,28)(19,30,31,37)(20,34,36,35)(24,39,41,40)(25,43,44,32)(29,46,33,48)(38,47,42,45)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,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,c^b=ch,e^b=ei,g^b=gi

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, 112857);