#7514578

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	#35757	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2679 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,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, 7421774);