#6435545

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433040
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	#36624	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2284 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)(7,11)(10,12)(13,14)(15,22)(16,21)(17,26)(18,20)(19,24)(23,28)(25,27);(5,6)(7,10)(8,9)(11,12)(13,15,17,24,18,23,16,25)(14,19,21,22,20,27,26,28);(2,4)(6,9)(10,12)(15,23)(19,27)(22,28)(24,25);(5,7,8,11)(6,10,9,12)(14,20)(19,27)(21,26)(22,28);(13,16,18,17)(14,21,20,26)(15,24,23,25)(19,28,27,22)

Polycyclic presentation:

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

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, 6342741);