#6684765

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#42889	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2040 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fg,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6591961);