#7566037

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×#3265	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2174 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fg,d^2=fgi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,e^b=ei,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, 7473233);