#7515946

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×#555	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1807 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)(7,12)(8,10)(9,11)(13,14,20,26)(15,21,28,39)(16,31,33,44)(17,36,27,42)(18,25,19,24)(22,35,43,34)(23,32,38,37)(29,41,30,40);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,21)(16,32)(17,35)(18,30)(19,29)(20,28)(22,42)(23,44)(24,41)(25,40)(26,39)(27,34)(31,38)(33,37)(36,43);(5,8,9,7)(6,10,11,12)(13,16,20,33)(14,22,26,43)(15,27,28,17)(18,35,19,34)(21,38,39,23)(24,44,25,31)(29,37,30,32)(36,41,42,40);(2,4)(13,17,20,27)(14,23,26,38)(15,16,28,33)(18,32,19,37)(21,22,39,43)(24,42,25,36)(29,35,30,34)(31,40,44,41);(6,11)(10,12)(15,28)(16,33)(21,39)(22,43)(29,30)(31,44)(34,35)(40,41)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fi,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,d^c=di,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, 7423142);