#6453023

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×#3511	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3288 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)(7,10)(8,11)(9,12)(13,14)(15,23)(16,20)(17,21)(18,22)(19,24)(25,28)(26,27);(2,4)(5,7,9,8)(6,10,12,11)(13,15,18,26)(14,19,22,28)(16,25,17,24)(20,23,21,27);(5,7,9,8)(6,11,12,10)(15,24)(16,17)(19,23)(20,21)(25,26)(27,28);(5,8,9,7)(6,11,12,10)(13,14)(15,23)(16,20)(17,21)(18,22)(19,24)(25,28)(26,27);(13,16,18,17)(14,20,22,21)(15,25,26,24)(19,23,28,27)

Polycyclic presentation:

2^9|b^2=gi,c^2=g,d^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,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.

Relationships

maximal subgroup of	#72307495, #72307543, #72307607
maximal normal subgroup of	#72307495, #72307543, #72307607

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6360219);