#7518449

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

2057 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)(15,17)(16,21)(18,20)(19,24)(22,25)(23,27)(26,28);(1,2)(3,4)(6,9)(10,11)(14,15)(16,23)(18,25)(19,27)(21,28)(24,26);(2,4)(13,15,17,14)(16,21,24,19)(18,22,25,20)(23,27,26,28);(2,4)(13,16)(14,19)(15,21)(17,24)(18,26)(20,27)(22,28)(23,25);(5,7,8,12)(6,10,9,11)(14,15)(19,21)(20,22)(27,28)

Polycyclic presentation:

2^9|c^2=i,e^2=fi,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, 7425645);