#6817084

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2034 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)(6,9)(7,11)(13,14)(15,17)(16,18)(19,24)(20,22)(21,23)(25,26)(27,28);(5,6)(7,10)(8,9)(11,12)(13,15,19,22)(14,20,24,17)(16,21,23,18)(25,27,28,26);(1,2,3,4)(6,9)(10,12)(13,16,20,26,19,23,17,27)(14,21,22,28,24,18,15,25);(5,7,8,11)(6,10,9,12)(13,17,19,20)(14,22,24,15)(16,26,23,27)(18,28,21,25);(13,18,20,25,19,21,17,28)(14,23,22,27,24,16,15,26)

Polycyclic presentation:

2^9|b^2=i,c^2=fgh,d^2=f,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6724280);