#6552779

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
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	#38910	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2005 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)(15,23)(16,18)(17,21)(19,22)(20,27)(24,26)(25,28);(5,6)(7,10)(8,9)(11,12)(13,15,17,25,19,26,18,27)(14,20,16,24,22,28,21,23);(2,4)(6,9)(10,12)(13,16,18,14,19,21,17,22)(15,24,27,20,26,23,25,28);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,21,22,16)(15,25,26,27)(20,23,28,24);(7,11)(10,12)(15,26)(20,28)(23,24)(25,27)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6459975);