#6380672

order	512
nilpotency class	3
isoclinism	isoclinic class of #61071
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2040 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)(8,12)(9,11)(13,14,19,24,18,21,17,23)(15,25,29,20,28,36,27,34)(16,22,33,40,31,37,32,39)(26,38,42,35,41,44,30,43);(5,7,9,8)(6,10,11,12)(13,15,18,28)(14,20,21,34)(16,30,31,42)(17,29,19,27)(22,38,37,44)(23,36,24,25)(26,33,41,32)(35,40,43,39);(5,8,9,7)(6,10,11,12)(14,21)(20,34)(22,37)(23,24)(25,36)(35,43)(38,44)(39,40);(13,16,17,32,18,31,19,33)(14,22,23,39,21,37,24,40)(15,26,27,30,28,41,29,42)(20,35,25,38,34,43,36,44);(16,31)(22,37)(26,41)(30,42)(32,33)(35,43)(38,44)(39,40)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gi,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,d^b=df,f^b=fi,h^b=hi,e^d=ei

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6287868);