#6447656

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437764
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3054 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=gi,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, 6354852);