#6433270

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433044
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2106 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14,19,24)(15,25,29,35)(16,30,31,38)(17,23,18,22)(20,28,36,27)(21,32,37,33)(26,34,40,44)(39,41,43,42);(5,7)(6,9)(8,11)(10,12)(13,15,18,28,19,29,17,27)(14,20,23,35,24,36,22,25)(16,26,33,42,31,40,32,41)(21,34,30,43,37,44,38,39);(2,4)(7,11)(9,12)(13,16,17,32,19,31,18,33)(14,21,22,38,24,37,23,30)(15,26,27,41,29,40,28,42)(20,34,25,39,36,44,35,43);(6,10)(9,12);(16,31)(21,37)(26,40)(30,38)(32,33)(34,44)(39,43)(41,42)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,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, 6340466);