#7518454

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	#35771	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2011 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(13,14)(15,19)(16,24)(17,23)(18,21)(20,26)(22,25)(27,28);(1,2)(3,4)(6,9)(10,11)(13,15,18,23)(14,17,21,19)(16,20,26,24)(22,27,28,25);(2,4)(13,15)(14,17)(16,22)(18,23)(19,21)(20,25)(24,27)(26,28);(2,4)(13,16,18,26)(14,20,21,24)(15,22,23,28)(17,25,19,27);(5,7,8,12)(6,10,9,11)(13,17,18,19)(14,15,21,23)(16,25,26,27)(20,22,24,28)

Polycyclic presentation:

2^9|b^2=i,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,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, 7425650);