#6542605

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#37845	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3019 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)(13,14,19,20,18,22,16,23)(15,24,26,17,21,27,25,28);(5,6)(7,10)(8,9)(11,12)(13,15,18,21)(14,17,22,28)(16,26,19,25)(20,24,23,27);(2,4)(6,9)(10,12)(13,16,18,19)(14,20,22,23)(15,25,21,26)(17,27,28,24);(5,7)(6,10)(8,11)(9,12)(13,17)(14,21)(15,22)(16,24)(18,28)(19,27)(20,25)(23,26);(7,11)(10,12)(14,22)(17,28)(20,23)(24,27)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,f^b=fi,d^c=di,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6449801);