#6439919

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2296 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:

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

Polycyclic presentation:

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

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, 6347115);