#6454638

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437764
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3027 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)(13,14,19,24)(15,25,30,43)(16,21,31,39)(17,22,33,40)(18,34,26,23)(20,35,38,29)(27,37,44,41)(28,42,32,36);(1,2)(3,4)(13,15,26,35,19,30,18,29)(14,20,34,43,24,38,23,25)(16,27,33,28,31,44,17,32)(21,36,22,41,39,42,40,37);(2,4)(6,9)(10,12)(13,16,19,31)(14,21,24,39)(15,27,30,44)(17,26,33,18)(20,36,38,42)(22,23,40,34)(25,37,43,41)(28,29,32,35);(2,4)(13,17)(14,22)(15,28)(16,26)(18,31)(19,33)(20,37)(21,23)(24,40)(25,42)(27,29)(30,32)(34,39)(35,44)(36,43)(38,41);(5,7,8,11)(6,10,9,12)(16,31)(17,33)(21,39)(22,40)(27,44)(28,32)(36,42)(37,41)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei

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, 6361834);