#7565970

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×#3284	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2043 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)(13,14,18,22)(15,19,26,34)(16,20,28,36)(17,21,30,38)(23,31,27,35)(24,32,25,33)(29,37,41,44)(39,42,40,43);(2,3)(6,10)(11,12)(13,15,18,26)(14,19,22,34)(16,27,28,23)(17,24,30,25)(20,35,36,31)(21,32,38,33)(29,40,41,39)(37,43,44,42);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,16)(14,20)(15,23)(17,29)(18,28)(19,31)(21,37)(22,36)(24,39)(25,40)(26,27)(30,41)(32,42)(33,43)(34,35)(38,44);(1,3,4,2)(5,7,9,8)(6,11,10,12)(15,24)(17,30)(19,32)(21,38)(23,39)(25,26)(27,40)(29,41)(31,42)(33,34)(35,43)(37,44);(5,8,9,7)(6,11,10,12)(16,28)(20,36)(23,27)(29,41)(31,35)(37,44)(39,40)(42,43)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fg,d^2=fg,e^2=f,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, 7473166);