#7520174

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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1821 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)(8,12)(9,10)(13,14)(15,25)(16,29)(17,32)(18,23)(19,36)(20,27)(21,31)(22,30)(24,35)(26,44)(28,40)(33,42)(34,41)(37,43)(38,39);(1,2)(3,4)(6,10)(11,12)(13,15,18,27)(14,20,23,25)(16,30,26,38)(17,33,34,31)(19,37,35,28)(21,41,42,32)(22,29,39,44)(24,43,36,40);(5,7,9,8)(6,11,10,12)(13,16)(14,21)(15,17)(18,26)(19,33)(20,39)(22,25)(23,42)(24,44)(27,34)(28,38)(29,36)(30,37)(31,35)(32,40)(41,43);(2,4)(13,17,18,34)(14,22,23,39)(15,26,27,16)(19,38,35,30)(20,21,25,42)(24,41,36,32)(28,31,37,33)(29,40,44,43);(5,8,9,7)(6,11,10,12)(14,23)(15,27)(16,26)(22,39)(24,36)(28,37)(31,33)(32,41)

Polycyclic presentation:

2^9|b^2=i,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=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, 7427370);