#7551439

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#45134	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2118 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7458635);