#201639

order	512
nilpotency class	3
isoclinism	isoclinic class of #201424
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2617	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1152 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)(19,29)(20,33,35,25)(21,27)(22,26)(23,28)(24,31)(30,43,46,48)(32,41)(34,40,37,42)(36,39)(38,45,47,44);(1,3)(2,7,9,15)(4,12)(5,13)(6,11)(8,10,14,16)(17,19,22,31,23,32,21,36)(18,24,27,29,28,39,26,41)(20,30,37,44,35,46,34,45)(25,38,42,43,33,47,40,48);(17,20)(18,25)(19,30)(21,34)(22,37)(23,35)(24,38)(26,40)(27,42)(28,33)(29,43)(31,44)(32,46)(36,45)(39,47)(41,48);(3,11)(7,15)(10,16)(12,13)(17,21,23,22)(18,26,28,27)(19,31,32,36)(20,34,35,37)(24,29,39,41)(25,40,33,42)(30,44,46,45)(38,43,47,48)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=i,e^2=h,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=dh,f^b=fi,e^c=ei,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318616, #72318627, #72318817
maximal normal subgroup of	#72318616, #72318627, #72318817

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108835);