#6467646

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#25797	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2268 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=hi,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,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, 6374842);