#6456361

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
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
prime radical	C₂×#3171	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2001 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6363557);