#6380501

order	512
nilpotency class	3
isoclinism	isoclinic class of #61049
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×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2192 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)(15,25)(16,22)(17,24)(18,23)(19,21)(20,27)(26,41)(28,39)(29,44)(30,40)(31,36)(32,34)(33,38)(35,42)(37,43);(5,7,9,8)(6,10,11,12)(13,15,19,28)(14,20,21,34)(16,29,31,42)(17,27,18,32)(22,37,36,41)(23,25,24,39)(26,33,43,30)(35,40,44,38);(5,8,9,7)(6,10,11,12)(14,21)(20,34)(22,36)(23,24)(25,39)(35,44)(37,41)(38,40);(13,16)(14,22)(15,26)(17,30)(18,33)(19,31)(20,35)(21,36)(23,38)(24,40)(25,41)(27,42)(28,43)(29,32)(34,44)(37,39);(13,17,19,18)(14,23,21,24)(15,27,28,32)(16,30,31,33)(20,25,34,39)(22,38,36,40)(26,42,43,29)(35,41,44,37)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6287697);