#6552680

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
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	#38033	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1873 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=fi,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=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, 6459876);