#7520809

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

1794 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=f,d^2=i,e^2=f,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, 7428005);