#7523085

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1775 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)(15,20)(16,26)(17,21)(18,22)(19,23)(24,34)(25,35)(27,43)(28,40)(29,36)(30,37)(31,38)(32,39)(33,41)(42,44);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15)(14,20)(16,18)(17,29)(19,28)(21,36)(22,26)(23,40)(24,31)(25,32)(27,33)(30,42)(34,38)(35,39)(37,44)(41,43);(2,4)(13,16,18,15)(14,20,22,26)(17,27,30,24)(19,28,32,25)(21,34,37,43)(23,35,39,40)(29,31,42,33)(36,41,44,38);(2,4)(5,7,9,8)(6,11,12,10)(13,17,18,30)(14,21,22,37)(15,24,16,27)(19,33,32,31)(20,34,26,43)(23,41,39,38)(25,42,28,29)(35,44,40,36);(5,8,9,7)(6,11,12,10)(15,16)(20,26)(24,27)(25,28)(29,42)(34,43)(35,40)(36,44)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,e^b=ei,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, 7430281);