#7565975

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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×#3285	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2002 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fgi,d^2=fg,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7473171);