#6439563

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2531 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,g^c=gi,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, 6346759);