#6440230

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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	#23871	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2156 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)(13,14,19,24)(15,20,17,22)(16,27,29,40)(18,32,34,23)(21,33,37,28)(25,41,30,44)(26,39,31,36)(35,43,38,42);(1,2)(3,4)(6,9)(10,12)(13,15,19,17)(14,20,24,22)(16,25,29,30)(18,26,34,31)(21,35,37,38)(23,36,32,39)(27,41,40,44)(28,42,33,43);(2,4)(13,16,19,29)(14,21,24,37)(15,25,17,30)(18,33,34,28)(20,35,22,38)(23,40,32,27)(26,43,31,42)(36,44,39,41);(2,4)(13,17,19,15)(14,22,24,20)(16,25,29,30)(18,31,34,26)(21,35,37,38)(23,39,32,36)(27,41,40,44)(28,42,33,43);(5,7,8,11)(6,10,9,12)(15,17)(20,22)(25,30)(26,31)(35,38)(36,39)(41,44)(42,43)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di,g^c=gi,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, 6347426);