#6441653

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2207 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 6348849);