#6448026

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437521
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2124 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6355222);