#7538090

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	C₂×#459	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1872 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)(5,6,8,10)(7,11,12,9)(13,14)(15,23)(16,20)(17,21)(18,22)(19,26)(24,35)(25,34)(27,33)(28,32)(29,36)(30,37)(31,38)(39,43)(40,42)(41,44);(1,2,4,3)(7,12)(9,11)(13,15)(14,19)(16,28)(17,26)(18,25)(20,35)(21,23)(22,33)(24,29)(27,31)(30,40)(32,36)(34,38)(37,43)(39,41)(42,44);(5,7,8,12)(6,9,10,11)(13,16,17,29)(14,20,21,36)(15,24,26,28)(18,30,31,41)(19,32,23,35)(22,37,38,44)(25,39,27,40)(33,42,34,43);(6,10)(9,11)(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=g,b^2=fi,c^2=gi,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, 7445286);