#7522988

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1757 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=f,e^2=fi,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, 7430184);