#6447200

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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	#35518	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2256 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=gi,c^2=fg,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6354396);