#6451812

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

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

4159 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:

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

Polycyclic presentation:

2^9|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=dhi,f^b=fi,h^b=hi,f^c=fi,h^c=hi,f^d=fi,h^d=hi

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, 6359008);