#6443694

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2373 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=gi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,e^d=ei

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, 6350890);