#7566028

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	#45057	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2086 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)(13,14,18,19)(15,20,16,23)(17,21,26,27)(22,28,24,25);(2,3)(6,10)(11,12)(13,15)(14,20)(16,18)(17,25)(19,23)(21,22)(24,27)(26,28);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,16,18,15)(14,20,19,23)(17,24,26,22)(21,28,27,25);(1,3,4,2)(5,7,9,8)(6,11,10,12)(13,17)(14,21)(15,22)(16,24)(18,26)(19,27)(20,28)(23,25);(5,8,9,7)(6,11,10,12)(15,16)(20,23)(22,24)(25,28)

Polycyclic presentation:

2^9|a^2=fi,c^2=fg,d^2=fgi,e^2=f,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, 7473224);