#6449929

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

1976 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)(7,10)(8,9)(11,12)(13,14,20,34,19,23,18,24)(15,25,33,21,29,38,28,36)(16,30,32,37,17,22,31,44)(26,40,42,43,27,39,41,35);(1,2)(3,4)(6,9)(10,12)(13,15,19,29)(14,21,23,36)(16,26,17,27)(18,33,20,28)(22,35,30,43)(24,38,34,25)(31,42,32,41)(37,40,44,39);(2,4)(6,9)(10,12)(13,16,19,17)(14,22,23,30)(15,26,29,27)(18,31,20,32)(21,35,36,43)(24,37,34,44)(25,39,38,40)(28,41,33,42);(2,4)(5,7,8,11)(6,10,9,12)(13,17,19,16)(14,22,23,30)(15,27,29,26)(18,32,20,31)(21,35,36,43)(24,37,34,44)(25,39,38,40)(28,42,33,41);(14,23)(16,17)(21,36)(24,34)(25,38)(26,27)(31,32)(41,42)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6357125);