#6456609

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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	#2433	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1974 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)(8,9)(11,12)(14,20)(15,23)(22,28)(25,26);(2,3)(13,14)(15,24)(16,25)(17,22)(18,20)(19,28)(21,26)(23,27);(1,2,4,3)(6,9)(10,12)(13,15)(14,21)(16,20)(17,26)(18,23)(19,25)(22,27)(24,28);(1,3,4,2)(13,16,18,21)(14,15,20,23)(17,27,19,24)(22,25,28,26);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,22,20,28)(15,25,23,26)(16,27,21,24)

Polycyclic presentation:

2^9|a^2=fi,c^2=fi,d^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,h^b=hi,e^c=ei,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, 6363805);