#7560544

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	#45412	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1850 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)(7,10)(8,9)(11,12)(13,14,19,24)(15,20,28,35)(16,21,29,36)(17,22,18,23)(25,32,41,44)(26,33,27,34)(30,37,31,38)(39,42,40,43);(2,3)(6,9)(10,12)(13,15,19,28)(14,20,24,35)(16,25,29,41)(17,27,18,26)(21,32,36,44)(22,34,23,33)(30,40,31,39)(37,43,38,42);(1,2,4,3)(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,25)(17,31)(18,30)(19,29)(20,32)(22,38)(23,37)(24,36)(26,40)(27,39)(28,41)(33,43)(34,42)(35,44);(1,3,4,2)(15,26)(16,29)(17,18)(20,33)(21,36)(22,23)(25,39)(27,28)(32,42)(34,35)(40,41)(43,44);(6,9)(10,12)(13,17,19,18)(14,22,24,23)(15,27,28,26)(16,30,29,31)(20,34,35,33)(21,37,36,38)(25,40,41,39)(32,43,44,42)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,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

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7467740);