#6552716

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	#38034	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1890 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)(13,14,18,22)(15,23,26,27)(16,19,28,25)(17,20,24,21);(5,6)(7,10)(8,9)(11,12)(13,15,24,19,18,26,17,25)(14,16,20,27,22,28,21,23);(2,4)(6,9)(10,12)(13,15,17,25,18,26,24,19)(14,16,21,23,22,28,20,27);(5,7)(6,10)(8,11)(9,12)(13,16,18,28)(14,19,22,25)(15,20,26,21)(17,23,24,27);(7,11)(10,12)(13,17,18,24)(14,20,22,21)(15,25,26,19)(16,27,28,23)

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=f,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^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, 6459912);