#205076

order	512
nilpotency class	3
isoclinism	isoclinic class of #19328
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#390
prime radical	C₂⁴	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

931 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)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(17,18)(19,29)(20,34)(21,26)(22,27)(23,28)(24,32)(25,36)(30,38)(31,40)(33,39)(35,42)(37,41)(43,44)(45,47)(46,48);(1,3,4,12,5,11,13,10)(2,6,8,16,7,14,15,9)(17,19,21,32,23,33,22,31)(18,24,27,29,28,40,26,39)(20,30,35,45,37,46,36,44)(25,38,41,43,42,48,34,47);(17,20,21,35,23,37,22,36)(18,25,27,41,28,42,26,34)(19,30,32,45,33,46,31,44)(24,38,29,43,40,48,39,47);(2,7)(6,14)(8,15)(9,16)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,37,36)(24,29,40,39)(25,34,42,41)(30,44,46,45)(38,43,48,47)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=fh,d^2=h,e^2=hi,f^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,d^b=dh,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, 112272);