#417140

order	512
nilpotency class	3
isoclinism	isoclinic class of #394799
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	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1396 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

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, 324336);