#6380504

order	512
nilpotency class	3
isoclinism	isoclinic class of #61058
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2118 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)(15,23)(16,20)(17,30)(18,21)(19,26)(22,33)(24,39)(25,41)(27,44)(28,40)(29,37)(31,34)(32,38)(35,43)(36,42);(5,7,9,8)(6,10,11,12)(13,15)(14,19)(16,27)(17,31)(18,24)(20,36)(21,34)(22,39)(23,30)(25,28)(26,33)(29,43)(32,42)(35,38)(37,41)(40,44);(5,8,9,7)(6,10,11,12)(15,24)(19,34)(23,39)(25,42)(26,31)(27,43)(35,44)(36,41);(13,16,18,29)(14,20,21,37)(15,25,24,42)(17,32,33,28)(19,35,34,44)(22,40,30,38)(23,41,39,36)(26,43,31,27);(14,21)(19,34)(20,37)(22,30)(23,39)(35,44)(36,41)(38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6287700);