#205544

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	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

928 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)(19,24,30,40)(20,32)(21,26)(22,37)(23,27)(25,35)(28,43,33,47)(29,39,31,46)(34,42)(36,48)(38,44,41,45);(1,3,4,11,5,10,13,16)(2,6,8,9,7,14,12,15)(17,19)(18,24,27,40)(20,33)(21,29)(22,31)(23,30)(25,41,42,38)(26,39,37,46)(28,36)(32,47,48,43)(34,45)(35,44);(2,7)(3,10)(8,12)(11,16)(17,20,21,34,23,36,22,35)(18,25,37,48,27,42,26,32)(19,28,29,44,30,33,31,45)(24,38,46,47,40,41,39,43);(17,21,23,22)(18,26,27,37)(19,29,30,31)(20,34,36,35)(24,39,40,46)(25,32,42,48)(28,44,33,45)(38,43,41,47)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=fh,d^2=i,e^2=hi,f^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, 112740);