#7518689

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	#24362	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2001 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|c^2=i,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,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, 7425885);