#413810

order	512
nilpotency class	3
isoclinism	isoclinic class of #395169
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1166 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=hi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 321006);