#7545983

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1738 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)(5,6,9,7)(8,11,12,10)(13,14)(15,26)(16,22)(17,23)(18,34)(19,25)(20,24)(21,30)(27,42)(28,43)(29,38)(31,41)(32,36)(33,40)(35,39)(37,44);(1,2,4,3)(5,7,9,6)(8,10,12,11)(13,15)(14,21)(16,27)(17,32)(18,29)(19,30)(20,35)(22,36)(23,42)(24,38)(25,26)(28,33)(31,44)(34,39)(37,40)(41,43);(5,8,9,12)(6,10,7,11)(13,16,19,17)(14,22,25,23)(15,27,30,32)(18,33,20,31)(21,36,26,42)(24,41,34,40)(28,35,44,29)(37,38,43,39);(6,7)(10,11)(13,17,19,16)(14,23,25,22)(15,28,30,44)(18,33,20,31)(21,37,26,43)(24,41,34,40)(27,35,32,29)(36,38,42,39);(1,3,4,2)(13,18,19,20)(14,24,25,34)(15,29,30,35)(16,31,17,33)(21,38,26,39)(22,40,23,41)(27,44,32,28)(36,37,42,43)

Polycyclic presentation:

2^9|a^2=g,b^2=fgi,c^2=gi,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=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, 7453179);