#6465931

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1906 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)(13,14,17,19)(15,23,24,39)(16,21,28,33)(18,31,25,22)(20,34,32,27)(26,38,29,42)(30,44,40,37)(35,43,36,41);(2,3)(6,9)(10,12)(14,19)(15,24)(16,28)(22,31)(27,34)(30,40)(35,36)(38,42);(1,2,4,3)(13,15,25,34,17,24,18,27)(14,20,31,39,19,32,22,23)(16,29,40,41,28,26,30,43)(21,36,44,38,33,35,37,42);(1,3,4,2)(13,16,17,28)(14,21,19,33)(15,26,24,29)(18,30,25,40)(20,35,32,36)(22,37,31,44)(23,38,39,42)(27,41,34,43);(5,7,8,11)(6,10,9,12)(16,28)(21,33)(26,29)(30,40)(35,36)(37,44)(38,42)(41,43)

Polycyclic presentation:

2^9|a^2=i,c^2=gh,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,d^c=di,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, 6373127);