#7561749

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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	#45136	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2163 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)(7,11)(8,12)(9,10)(14,19)(15,23)(16,27)(22,32)(26,39)(30,33)(35,37)(41,44);(2,3)(6,10)(11,12)(13,14,17,19)(15,24,23,20)(16,28,27,38)(18,32,31,22)(21,33,34,30)(25,41,29,44)(26,36,39,40)(35,42,37,43);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,15,17,23)(14,20,19,24)(16,29,27,25)(18,26,31,39)(21,37,34,35)(22,36,32,40)(28,41,38,44)(30,43,33,42);(1,3,4,2)(5,8,9,7)(6,12,10,11)(13,16,17,27)(14,21,19,34)(15,25,23,29)(18,33,31,30)(20,35,24,37)(22,38,32,28)(26,43,39,42)(36,41,40,44);(6,10)(11,12)(15,23)(16,27)(20,24)(21,34)(26,39)(28,38)(30,33)(36,40)

Polycyclic presentation:

2^9|b^2=i,c^2=fgi,d^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7468945);