#7564786

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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	#25761	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1904 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)(15,19)(16,20)(17,22)(18,21)(23,30)(24,36)(25,32)(26,31)(27,33)(28,34)(29,35)(37,44)(38,41)(39,42)(40,43);(2,3)(6,10)(11,12)(13,15,18,25)(14,19,21,32)(16,23,27,39)(17,24,29,26)(20,30,33,42)(22,36,35,31)(28,37,40,38)(34,44,43,41);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,16)(14,20)(15,23)(17,28)(18,27)(19,30)(21,33)(22,34)(24,37)(25,39)(26,38)(29,40)(31,41)(32,42)(35,43)(36,44);(1,3,4,2)(14,21)(15,24)(17,29)(19,31)(20,33)(23,37)(25,26)(28,40)(30,41)(32,36)(38,39)(42,44);(5,8,9,7)(6,11,10,12)(15,25)(16,27)(19,32)(20,33)(24,26)(28,40)(31,36)(34,43)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7471982);