#454913

order	512
nilpotency class	3
isoclinism	isoclinic class of #453996
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	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1487 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 362109);