#6466215

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2198 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)(7,11)(8,9)(10,12)(13,14,18,23,20,25,19,24)(15,26,28,36,29,35,16,21)(17,22,32,39,31,38,33,40)(27,41,43,44,42,37,30,34);(2,4)(6,9)(10,11)(13,15,20,29)(14,21,25,36)(16,19,28,18)(17,30,31,43)(22,37,38,41)(23,35,24,26)(27,32,42,33)(34,39,44,40);(5,7,8,12)(6,10,9,11)(13,16,19,29,20,28,18,15)(14,21,24,35,25,36,23,26)(17,30,33,42,31,43,32,27)(22,34,40,37,38,44,39,41);(13,17,19,33,20,31,18,32)(14,22,24,40,25,38,23,39)(15,27,16,30,29,42,28,43)(21,34,35,37,36,44,26,41);(17,31)(22,38)(27,42)(30,43)(32,33)(34,44)(37,41)(39,40)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=gh,d^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,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, 6373411);