#7520522

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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	#45316	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1878 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)(15,25)(16,29)(17,32)(18,23)(19,37)(20,27)(21,31)(22,35)(24,36)(26,43)(28,40)(30,39)(33,41)(34,44)(38,42);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15)(14,20)(16,30)(17,33)(18,27)(19,38)(21,32)(22,29)(23,25)(24,42)(26,35)(28,36)(31,34)(37,40)(39,43)(41,44);(5,7,9,8)(6,11,12,10)(13,16)(14,21)(15,17)(18,26)(19,33)(20,39)(22,25)(23,41)(24,43)(27,34)(28,30)(29,37)(31,36)(32,40)(35,38)(42,44);(2,4)(13,17,18,34)(14,22,23,39)(15,26,27,16)(19,30,36,35)(20,21,25,41)(24,44,37,32)(28,31,38,33)(29,40,43,42);(5,8,9,7)(6,11,12,10)(14,23)(15,27)(16,26)(22,39)(24,37)(28,38)(31,33)(32,44)

Polycyclic presentation:

2^9|b^2=fi,c^2=fi,d^2=i,e^2=fi,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=cgi,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

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7427718);