#7557496

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1766 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,d^2=g,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,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, 7464692);