#417142

order	512
nilpotency class	3
isoclinism	isoclinic class of #394801
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

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

1350 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=fg,d^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=di,e^c=ei,f^c=fi,g^c=gi

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, 324338);