#6440453

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3499	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4174 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,h^a=hi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,f^d=fi,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, 6347649);