#6452213

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	#35518	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3333 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,d^2=g,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,e^b=ei,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, 6359409);