#6444613

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2088 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fgi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dhi,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351809);