#7515370

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	#23853	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1970 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)(7,10)(8,9)(11,12)(13,14,17,22)(15,24,27,19)(16,29,31,38)(18,20,33,23)(21,34,39,32)(25,40,28,36)(26,42,44,37)(30,41,43,35);(1,2)(3,4)(6,9)(10,12)(13,15,17,27)(14,19,22,24)(16,30,31,43)(18,25,33,28)(20,36,23,40)(21,37,39,42)(26,34,44,32)(29,35,38,41);(5,7,8,11)(6,10,9,12)(14,20)(15,25)(16,31)(18,33)(21,38)(22,23)(26,43)(27,28)(29,39)(30,44)(35,41)(36,40);(2,4)(13,16,17,31)(14,21,22,39)(15,26,27,44)(18,34,33,32)(19,35,24,41)(20,29,23,38)(25,30,28,43)(36,42,40,37);(6,9)(10,12)(14,22)(15,27)(16,31)(20,23)(25,28)(32,34)(35,41)(37,42)

Polycyclic presentation:

2^9|a^2=i,b^2=i,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,g^c=gi,h^c=hi,e^d=ei,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, 7422566);