#6443690

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	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2330 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,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, 6350886);