#205587

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	#60673	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

966 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,d^2=i,e^2=hi,f^2=i,g^2=h,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.

Relationships

maximal subgroup of	#72237262, #72237264, #72237266, …
maximal normal subgroup of	#72237262, #72237264, #72237266, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112783);