#6464024

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437764
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	C₂²×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2082 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=g,d^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,e^c=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, 6371220);