#6447108

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
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×D₈	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2326 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,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=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6354304);