#7522389

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	#45241	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2173 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)(7,11)(8,10)(9,12)(13,14,18,22)(15,19,24,31)(16,20,27,34)(17,21,28,35)(23,30,38,42)(25,32,39,43)(26,33,29,36)(37,41,40,44);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(16,26)(17,28)(20,33)(21,35)(23,37)(25,39)(27,29)(30,41)(32,43)(34,36)(38,40)(42,44);(2,4)(13,15,18,24)(14,19,22,31)(16,23,27,38)(17,25,28,39)(20,30,34,42)(21,32,35,43)(26,37,29,40)(33,41,36,44);(2,4)(5,7,9,8)(6,11,12,10)(13,16,18,27)(14,20,22,34)(15,23,24,38)(17,29,28,26)(19,30,31,42)(21,36,35,33)(25,40,39,37)(32,44,43,41);(5,8,9,7)(6,11,12,10)(15,24)(19,31)(23,38)(25,39)(30,42)(32,43)(37,40)(41,44)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=fi,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, 7429585);