#6443788

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	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2170 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)(7,12)(8,10)(9,11)(13,14,17,22)(15,20,25,19)(16,26,18,21)(23,24,28,27);(5,7,9,8)(6,10,11,12)(13,14)(15,23)(16,21)(17,22)(18,26)(19,24)(20,27)(25,28);(1,2,3,4)(13,15,17,25)(14,19,22,20)(16,24,18,27)(21,28,26,23);(5,8,9,7)(6,10,11,12)(13,15,17,25)(14,20,22,19)(16,24,18,27)(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=i,b^2=fi,c^2=g,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, 6350984);