#6400817

order	512
nilpotency class	3
isoclinism	isoclinic class of #61643
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1708 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6308013);